Եռանկյունաչափական ֆունկցիաների ինտեգրալների ցանկ

Եռանկյունաչափական ֆունկցիաների ինտեգրալների ցանկ, ստորև ներկայացված են սինուս, կոսինուս, տանգենս, կոտանգենս, սեկանս, կոսեկանս ֆունկցիաների ինտեգրալների ցանկերը։ Անորոշ ինտեգրալների համար ինտեգրման հաստատունը բաց է թողնված։


== Սինուս ==

  
    
      
        ∫
        sin
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            1
            c
          
        
        cos
        ⁡
        c
        x
        
        
      
    
    {\displaystyle \int \sin cx\;dx=-{\frac {1}{c}}\cos cx\,\!}
  
  
    
      
        ∫
        
          sin
          
            n
          
        
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
              cos
              ⁡
              c
              x
            
            
              n
              c
            
          
        
        +
        
          
            
              n
              −
              1
            
            n
          
        
        ∫
        
          sin
          
            n
            −
            2
          
        
        ⁡
        c
        x
        
        d
        x
        
        
          
            ( 
          
        
        n
        >
        0
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sin ^{n}cx\;dx=-{\frac {\sin ^{n-1}cx\cos cx}{nc}}+{\frac {n-1}{n}}\int \sin ^{n-2}cx\;dx\qquad {\mbox{( }}n>0{\mbox{)}}\,\!}
  
  
    
      
        ∫
        x
        sin
        ⁡
        c
        x
        
        d
        x
        =
        
          
            
              sin
              ⁡
              c
              x
            
            
              c
              
                2
              
            
          
        
        −
        
          
            
              x
              cos
              ⁡
              c
              x
            
            c
          
        
        
        
      
    
    {\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}}}-{\frac {x\cos cx}{c}}\,\!}
  
  
    
      
        ∫
        
          x
          
            2
          
        
        sin
        ⁡
        c
        x
        
        d
        x
        =
        
          
            
              2
              cos
              ⁡
              c
              x
            
            
              c
              
                3
              
            
          
        
        +
        
          
            
              2
              x
              sin
              ⁡
              c
              x
            
            
              c
              
                2
              
            
          
        
        −
        
          
            
              
                x
                
                  2
                
              
              cos
              ⁡
              c
              x
            
            c
          
        
        
        
      
    
    {\displaystyle \int x^{2}\sin cx\;dx={\frac {2\cos cx}{c^{3}}}+{\frac {2x\sin cx}{c^{2}}}-{\frac {x^{2}\cos cx}{c}}\,\!}
  
  
    
      
        ∫
        
          x
          
            3
          
        
        sin
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            
              6
              sin
              ⁡
              c
              x
            
            
              c
              
                4
              
            
          
        
        +
        
          
            
              6
              x
              cos
              ⁡
              c
              x
            
            
              c
              
                3
              
            
          
        
        +
        
          
            
              3
              
                x
                
                  2
                
              
              sin
              ⁡
              c
              x
            
            
              c
              
                2
              
            
          
        
        −
        
          
            
              
                x
                
                  3
                
              
              cos
              ⁡
              c
              x
            
            c
          
        
        
        
      
    
    {\displaystyle \int x^{3}\sin cx\;dx=-{\frac {6\sin cx}{c^{4}}}+{\frac {6x\cos cx}{c^{3}}}+{\frac {3x^{2}\sin cx}{c^{2}}}-{\frac {x^{3}\cos cx}{c}}\,\!}
  
  
    
      
        ∫
        
          x
          
            4
          
        
        sin
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            
              24
              cos
              ⁡
              c
              x
            
            
              c
              
                5
              
            
          
        
        −
        
          
            
              24
              x
              sin
              ⁡
              c
              x
            
            
              c
              
                4
              
            
          
        
        +
        
          
            
              12
              
                x
                
                  2
                
              
              cos
              ⁡
              c
              x
            
            
              c
              
                3
              
            
          
        
        +
        
          
            
              4
              
                x
                
                  3
                
              
              sin
              ⁡
              c
              x
            
            
              c
              
                2
              
            
          
        
        −
        
          
            
              
                x
                
                  4
                
              
              cos
              ⁡
              c
              x
            
            c
          
        
        
        
      
    
    {\displaystyle \int x^{4}\sin cx\;dx=-{\frac {24\cos cx}{c^{5}}}-{\frac {24x\sin cx}{c^{4}}}+{\frac {12x^{2}\cos cx}{c^{3}}}+{\frac {4x^{3}\sin cx}{c^{2}}}-{\frac {x^{4}\cos cx}{c}}\,\!}
  
  
    
      
        ∫
        
          x
          
            5
          
        
        sin
        ⁡
        c
        x
        
        d
        x
        =
        
          
            
              120
              sin
              ⁡
              c
              x
            
            
              c
              
                6
              
            
          
        
        −
        
          
            
              120
              x
              cos
              ⁡
              c
              x
            
            
              c
              
                5
              
            
          
        
        −
        
          
            
              60
              
                x
                
                  2
                
              
              sin
              ⁡
              c
              x
            
            
              c
              
                4
              
            
          
        
        +
        
          
            
              20
              
                x
                
                  3
                
              
              cos
              ⁡
              c
              x
            
            
              c
              
                3
              
            
          
        
        +
        
          
            
              5
              
                x
                
                  4
                
              
              sin
              ⁡
              c
              x
            
            
              c
              
                2
              
            
          
        
        −
        
          
            
              
                x
                
                  5
                
              
              cos
              ⁡
              c
              x
            
            c
          
        
        
        
      
    
    {\displaystyle \int x^{5}\sin cx\;dx={\frac {120\sin cx}{c^{6}}}-{\frac {120x\cos cx}{c^{5}}}-{\frac {60x^{2}\sin cx}{c^{4}}}+{\frac {20x^{3}\cos cx}{c^{3}}}+{\frac {5x^{4}\sin cx}{c^{2}}}-{\frac {x^{5}\cos cx}{c}}\,\!}
  
  
    
      
        
          
            
              
                ∫
                
                  x
                  
                    n
                  
                
                sin
                ⁡
                c
                x
                
                d
                x
              
              
                
                =
                n
                !
                ⋅
                sin
                ⁡
                c
                x
                
                  [
                  
                    
                      
                        
                          x
                          
                            n
                            −
                            1
                          
                        
                        
                          
                            c
                            
                              2
                            
                          
                          ⋅
                          (
                          n
                          −
                          1
                          )
                          !
                        
                      
                    
                    −
                    
                      
                        
                          x
                          
                            n
                            −
                            3
                          
                        
                        
                          
                            c
                            
                              4
                            
                          
                          ⋅
                          (
                          n
                          −
                          3
                          )
                          !
                        
                      
                    
                    +
                    
                      
                        
                          x
                          
                            n
                            −
                            5
                          
                        
                        
                          
                            c
                            
                              6
                            
                          
                          ⋅
                          (
                          n
                          −
                          5
                          )
                          !
                        
                      
                    
                    −
                    .
                    .
                    .
                  
                  ]
                
                −
              
            
            
              
              
                
                −
                n
                !
                ⋅
                cos
                ⁡
                c
                x
                
                  [
                  
                    
                      
                        
                          x
                          
                            n
                          
                        
                        
                          c
                          ⋅
                          n
                          !
                        
                      
                    
                    −
                    
                      
                        
                          x
                          
                            n
                            −
                            2
                          
                        
                        
                          
                            c
                            
                              3
                            
                          
                          ⋅
                          (
                          n
                          −
                          2
                          )
                          !
                        
                      
                    
                    +
                    
                      
                        
                          x
                          
                            n
                            −
                            4
                          
                        
                        
                          
                            c
                            
                              5
                            
                          
                          ⋅
                          (
                          n
                          −
                          4
                          )
                          !
                        
                      
                    
                    −
                    .
                    .
                    .
                  
                  ]
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\int x^{n}\sin cx\;dx&=n!\cdot \sin cx\left[{\frac {x^{n-1}}{c^{2}\cdot (n-1)!}}-{\frac {x^{n-3}}{c^{4}\cdot (n-3)!}}+{\frac {x^{n-5}}{c^{6}\cdot (n-5)!}}-...\right]-\\&-n!\cdot \cos cx\left[{\frac {x^{n}}{c\cdot n!}}-{\frac {x^{n-2}}{c^{3}\cdot (n-2)!}}+{\frac {x^{n-4}}{c^{5}\cdot (n-4)!}}-...\right]\end{aligned}}}
  
  
    
      
        ∫
        
          x
          
            n
          
        
        sin
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            
              x
              
                n
              
            
            c
          
        
        cos
        ⁡
        c
        x
        +
        
          
            n
            c
          
        
        ∫
        
          x
          
            n
            −
            1
          
        
        cos
        ⁡
        c
        x
        
        d
        x
        
        
          
            ( 
          
        
        n
        >
        0
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int x^{n}\sin cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad {\mbox{( }}n>0{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
            
            x
          
        
        d
        x
        =
        
          ∑
          
            i
            =
            0
          
          
            ∞
          
        
        (
        −
        1
        
          )
          
            i
          
        
        
          
            
              (
              c
              x
              
                )
                
                  2
                  i
                  +
                  1
                
              
            
            
              (
              2
              i
              +
              1
              )
              ⋅
              (
              2
              i
              +
              1
              )
              !
            
          
        
        
        
      
    
    {\displaystyle \int {\frac {\sin cx}{x}}dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}\,\!}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
            
            
              x
              
                n
              
            
          
        
        d
        x
        =
        −
        
          
            
              sin
              ⁡
              c
              x
            
            
              (
              n
              −
              1
              )
              
                x
                
                  n
                  −
                  1
                
              
            
          
        
        +
        
          
            c
            
              n
              −
              1
            
          
        
        ∫
        
          
            
              cos
              ⁡
              c
              x
            
            
              x
              
                n
                −
                1
              
            
          
        
        d
        x
        
        
      
    
    {\displaystyle \int {\frac {\sin cx}{x^{n}}}dx=-{\frac {\sin cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos cx}{x^{n-1}}}dx\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            c
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            
              
                
                  c
                  x
                
                2
              
            
          
          |
        
      
    
    {\displaystyle \int {\frac {dx}{\sin cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            
              cos
              ⁡
              c
              x
            
            
              c
              (
              1
              −
              n
              )
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        +
        
          
            
              n
              −
              2
            
            
              n
              −
              1
            
          
        
        ∫
        
          
            
              d
              x
            
            
              
                sin
                
                  n
                  −
                  2
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        n
        >
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {dx}{\sin ^{n}cx}}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad {\mbox{( }}n>1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              1
              ±
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            c
          
        
        tg
        ⁡
        
          (
          
            
              
                
                  c
                  x
                
                2
              
            
            ∓
            
              
                π
                4
              
            
          
          )
        
      
    
    {\displaystyle \int {\frac {dx}{1\pm \sin cx}}={\frac {1}{c}}\operatorname {tg} \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}
  
  
    
      
        ∫
        
          
            
              x
              
              d
              x
            
            
              1
              +
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            x
            c
          
        
        tg
        ⁡
        
          (
          
            
              
                
                  c
                  x
                
                2
              
            
            −
            
              
                π
                4
              
            
          
          )
        
        +
        
          
            2
            
              c
              
                2
              
            
          
        
        ln
        ⁡
        
          |
          
            cos
            ⁡
            
              (
              
                
                  
                    
                      c
                      x
                    
                    2
                  
                
                −
                
                  
                    π
                    4
                  
                
              
              )
            
          
          |
        
      
    
    {\displaystyle \int {\frac {x\;dx}{1+\sin cx}}={\frac {x}{c}}\operatorname {tg} \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{c^{2}}}\ln \left|\cos \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)\right|}
  
  
    
      
        ∫
        
          
            
              x
              
              d
              x
            
            
              1
              −
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            x
            c
          
        
        ctg
        ⁡
        
          (
          
            
              
                π
                4
              
            
            −
            
              
                
                  c
                  x
                
                2
              
            
          
          )
        
        +
        
          
            2
            
              c
              
                2
              
            
          
        
        ln
        ⁡
        
          |
          
            sin
            ⁡
            
              (
              
                
                  
                    π
                    4
                  
                
                −
                
                  
                    
                      c
                      x
                    
                    2
                  
                
              
              )
            
          
          |
        
      
    
    {\displaystyle \int {\frac {x\;dx}{1-\sin cx}}={\frac {x}{c}}\operatorname {ctg} \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
              
              d
              x
            
            
              1
              ±
              sin
              ⁡
              c
              x
            
          
        
        =
        ±
        x
        +
        
          
            1
            c
          
        
        tg
        ⁡
        
          (
          
            
              
                π
                4
              
            
            ∓
            
              
                
                  c
                  x
                
                2
              
            
          
          )
        
      
    
    {\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}}=\pm x+{\frac {1}{c}}\operatorname {tg} \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}
  
  
    
      
        ∫
        sin
        ⁡
        
          c
          
            1
          
        
        x
        sin
        ⁡
        
          c
          
            2
          
        
        x
        
        d
        x
        =
        
          
            
              sin
              ⁡
              (
              
                c
                
                  1
                
              
              −
              
                c
                
                  2
                
              
              )
              x
            
            
              2
              (
              
                c
                
                  1
                
              
              −
              
                c
                
                  2
                
              
              )
            
          
        
        −
        
          
            
              sin
              ⁡
              (
              
                c
                
                  1
                
              
              +
              
                c
                
                  2
                
              
              )
              x
            
            
              2
              (
              
                c
                
                  1
                
              
              +
              
                c
                
                  2
                
              
              )
            
          
        
        
        
          
            ( 
          
        
        
          |
        
        
          c
          
            1
          
        
        
          |
        
        ≠
        
          |
        
        
          c
          
            2
          
        
        
          |
        
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}
  


== Կոսինուս ==

  
    
      
        ∫
        cos
        ⁡
        a
        x
        
        
          d
        
        x
        =
        
          
            1
            a
          
        
        sin
        ⁡
        a
        x
        +
        C
        
        
      
    
    {\displaystyle \int \cos ax\;\mathrm {d} x={\frac {1}{a}}\sin ax+C\,\!}
  
  
    
      
        ∫
        
          cos
          
            2
          
        
        ⁡
        
          a
          x
        
        
        
          d
        
        x
        =
        
          
            x
            2
          
        
        +
        
          
            1
            
              4
              a
            
          
        
        sin
        ⁡
        2
        a
        x
        +
        C
        =
        
          
            x
            2
          
        
        +
        
          
            1
            
              2
              a
            
          
        
        sin
        ⁡
        a
        x
        cos
        ⁡
        a
        x
        +
        C
        
      
    
    {\displaystyle \int \cos ^{2}{ax}\;\mathrm {d} x={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}+{\frac {1}{2a}}\sin ax\cos ax+C\!}
  
  
    
      
        ∫
        
          cos
          
            n
          
        
        ⁡
        a
        x
        
        
          d
        
        x
        =
        
          
            
              
                cos
                
                  n
                  −
                  1
                
              
              ⁡
              a
              x
              sin
              ⁡
              a
              x
            
            
              n
              a
            
          
        
        +
        
          
            
              n
              −
              1
            
            n
          
        
        ∫
        
          cos
          
            n
            −
            2
          
        
        ⁡
        a
        x
        
        
          d
        
        x
        
        
          
            (for 
          
        
        n
        >
        0
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \cos ^{n}ax\;\mathrm {d} x={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}
  
  
    
      
        ∫
        x
        cos
        ⁡
        a
        x
        
        
          d
        
        x
        =
        
          
            
              cos
              ⁡
              a
              x
            
            
              a
              
                2
              
            
          
        
        +
        
          
            
              x
              sin
              ⁡
              a
              x
            
            a
          
        
        +
        C
        
        
      
    
    {\displaystyle \int x\cos ax\;\mathrm {d} x={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C\,\!}
  
  
    
      
        ∫
        
          x
          
            2
          
        
        
          cos
          
            2
          
        
        ⁡
        
          a
          x
        
        
        
          d
        
        x
        =
        
          
            
              x
              
                3
              
            
            6
          
        
        +
        
          (
          
            
              
                
                  x
                  
                    2
                  
                
                
                  4
                  a
                
              
            
            −
            
              
                1
                
                  8
                  
                    a
                    
                      3
                    
                  
                
              
            
          
          )
        
        sin
        ⁡
        2
        a
        x
        +
        
          
            x
            
              4
              
                a
                
                  2
                
              
            
          
        
        cos
        ⁡
        2
        a
        x
        +
        C
        
      
    
    {\displaystyle \int x^{2}\cos ^{2}{ax}\;\mathrm {d} x={\frac {x^{3}}{6}}+\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax+{\frac {x}{4a^{2}}}\cos 2ax+C\!}
  
  
    
      
        ∫
        
          x
          
            n
          
        
        cos
        ⁡
        a
        x
        
        
          d
        
        x
        =
        
          
            
              
                x
                
                  n
                
              
              sin
              ⁡
              a
              x
            
            a
          
        
        −
        
          
            n
            a
          
        
        ∫
        
          x
          
            n
            −
            1
          
        
        sin
        ⁡
        a
        x
        
        
          d
        
        x
        
        =
        
          ∑
          
            k
            =
            0
          
          
            2
            k
            +
            1
            ≤
            n
          
        
        (
        −
        1
        
          )
          
            k
          
        
        
          
            
              x
              
                n
                −
                2
                k
                −
                1
              
            
            
              a
              
                2
                +
                2
                k
              
            
          
        
        
          
            
              n
              !
            
            
              (
              n
              −
              2
              k
              −
              1
              )
              !
            
          
        
        cos
        ⁡
        a
        x
        +
        
          ∑
          
            k
            =
            0
          
          
            2
            k
            ≤
            n
          
        
        (
        −
        1
        
          )
          
            k
          
        
        
          
            
              x
              
                n
                −
                2
                k
              
            
            
              a
              
                1
                +
                2
                k
              
            
          
        
        
          
            
              n
              !
            
            
              (
              n
              −
              2
              k
              )
              !
            
          
        
        sin
        ⁡
        a
        x
        
      
    
    {\displaystyle \int x^{n}\cos ax\;\mathrm {d} x={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\;\mathrm {d} x\,=\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-2k-1}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\cos ax+\sum _{k=0}^{2k\leq n}(-1)^{k}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\sin ax\!}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              a
              x
            
            x
          
        
        
          d
        
        x
        =
        ln
        ⁡
        
          |
        
        a
        x
        
          |
        
        +
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        (
        −
        1
        
          )
          
            k
          
        
        
          
            
              (
              a
              x
              
                )
                
                  2
                  k
                
              
            
            
              2
              k
              ⋅
              (
              2
              k
              )
              !
            
          
        
        +
        C
        
        
      
    
    {\displaystyle \int {\frac {\cos ax}{x}}\mathrm {d} x=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C\,\!}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              a
              x
            
            
              x
              
                n
              
            
          
        
        
          d
        
        x
        =
        −
        
          
            
              cos
              ⁡
              a
              x
            
            
              (
              n
              −
              1
              )
              
                x
                
                  n
                  −
                  1
                
              
            
          
        
        −
        
          
            a
            
              n
              −
              1
            
          
        
        ∫
        
          
            
              sin
              ⁡
              a
              x
            
            
              x
              
                n
                −
                1
              
            
          
        
        
          d
        
        x
        
        
          
            (for 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\cos ax}{x^{n}}}\mathrm {d} x=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}\mathrm {d} x\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                d
              
              x
            
            
              cos
              ⁡
              a
              x
            
          
        
        =
        
          
            1
            a
          
        
        ln
        ⁡
        
          |
          
            tan
            ⁡
            
              (
              
                
                  
                    
                      a
                      x
                    
                    2
                  
                
                +
                
                  
                    π
                    4
                  
                
              
              )
            
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {\mathrm {d} x}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
  
  
    
      
        ∫
        
          
            
              
                d
              
              x
            
            
              
                cos
                
                  n
                
              
              ⁡
              a
              x
            
          
        
        =
        
          
            
              sin
              ⁡
              a
              x
            
            
              a
              (
              n
              −
              1
              )
              
                cos
                
                  n
                  −
                  1
                
              
              ⁡
              a
              x
            
          
        
        +
        
          
            
              n
              −
              2
            
            
              n
              −
              1
            
          
        
        ∫
        
          
            
              
                d
              
              x
            
            
              
                cos
                
                  n
                  −
                  2
                
              
              ⁡
              a
              x
            
          
        
        
        
          
            (for 
          
        
        n
        >
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\mathrm {d} x}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                d
              
              x
            
            
              1
              +
              cos
              ⁡
              a
              x
            
          
        
        =
        
          
            1
            a
          
        
        tan
        ⁡
        
          
            
              a
              x
            
            2
          
        
        +
        C
        
        
      
    
    {\displaystyle \int {\frac {\mathrm {d} x}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}
  
  
    
      
        ∫
        
          
            
              
                d
              
              x
            
            
              1
              −
              cos
              ⁡
              a
              x
            
          
        
        =
        −
        
          
            1
            a
          
        
        cot
        ⁡
        
          
            
              a
              x
            
            2
          
        
        +
        C
      
    
    {\displaystyle \int {\frac {\mathrm {d} x}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}
  
  
    
      
        ∫
        
          
            
              x
              
              
                d
              
              x
            
            
              1
              +
              cos
              ⁡
              a
              x
            
          
        
        =
        
          
            x
            a
          
        
        tan
        ⁡
        
          
            
              a
              x
            
            2
          
        
        +
        
          
            2
            
              a
              
                2
              
            
          
        
        ln
        ⁡
        
          |
          
            cos
            ⁡
            
              
                
                  a
                  x
                
                2
              
            
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {x\;\mathrm {d} x}{1+\cos ax}}={\frac {x}{a}}\tan {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\cos {\frac {ax}{2}}\right|+C}
  
  
    
      
        ∫
        
          
            
              x
              
              
                d
              
              x
            
            
              1
              −
              cos
              ⁡
              a
              x
            
          
        
        =
        −
        
          
            x
            a
          
        
        cot
        ⁡
        
          
            
              a
              x
            
            2
          
        
        +
        
          
            2
            
              a
              
                2
              
            
          
        
        ln
        ⁡
        
          |
          
            sin
            ⁡
            
              
                
                  a
                  x
                
                2
              
            
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {x\;\mathrm {d} x}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              a
              x
              
              
                d
              
              x
            
            
              1
              +
              cos
              ⁡
              a
              x
            
          
        
        =
        x
        −
        
          
            1
            a
          
        
        tan
        ⁡
        
          
            
              a
              x
            
            2
          
        
        +
        C
        
        
      
    
    {\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              a
              x
              
              
                d
              
              x
            
            
              1
              −
              cos
              ⁡
              a
              x
            
          
        
        =
        −
        x
        −
        
          
            1
            a
          
        
        cot
        ⁡
        
          
            
              a
              x
            
            2
          
        
        +
        C
        
        
      
    
    {\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}
  
  
    
      
        ∫
        cos
        ⁡
        
          a
          
            1
          
        
        x
        cos
        ⁡
        
          a
          
            2
          
        
        x
        
        
          d
        
        x
        =
        
          
            
              sin
              ⁡
              (
              
                a
                
                  2
                
              
              −
              
                a
                
                  1
                
              
              )
              x
            
            
              2
              (
              
                a
                
                  2
                
              
              −
              
                a
                
                  1
                
              
              )
            
          
        
        +
        
          
            
              sin
              ⁡
              (
              
                a
                
                  2
                
              
              +
              
                a
                
                  1
                
              
              )
              x
            
            
              2
              (
              
                a
                
                  2
                
              
              +
              
                a
                
                  1
                
              
              )
            
          
        
        +
        C
        
        
          
            (for 
          
        
        
          |
        
        
          a
          
            1
          
        
        
          |
        
        ≠
        
          |
        
        
          a
          
            2
          
        
        
          |
        
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \cos a_{1}x\cos a_{2}x\;\mathrm {d} x={\frac {\sin(a_{2}-a_{1})x}{2(a_{2}-a_{1})}}+{\frac {\sin(a_{2}+a_{1})x}{2(a_{2}+a_{1})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}
  


== Տանգենս ==

  
    
      
        ∫
        tg
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            1
            c
          
        
        ln
        ⁡
        
          |
        
        cos
        ⁡
        c
        x
        
          |
        
        
        
      
    
    {\displaystyle \int \operatorname {tg} cx\;dx=-{\frac {1}{c}}\ln |\cos cx|\,\!}
  
  
    
      
        ∫
        
          tg
          
            n
          
        
        ⁡
        c
        x
        
        d
        x
        =
        
          
            1
            
              c
              (
              n
              −
              1
              )
            
          
        
        
          tg
          
            n
            −
            1
          
        
        ⁡
        c
        x
        −
        ∫
        
          tg
          
            n
            −
            2
          
        
        ⁡
        c
        x
        
        d
        x
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \operatorname {tg} ^{n}cx\;dx={\frac {1}{c(n-1)}}\operatorname {tg} ^{n-1}cx-\int \operatorname {tg} ^{n-2}cx\;dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              tg
              ⁡
              c
              x
              +
              1
            
          
        
        =
        
          
            x
            2
          
        
        +
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
        
        sin
        ⁡
        c
        x
        +
        cos
        ⁡
        c
        x
        
          |
        
        
        
      
    
    {\displaystyle \int {\frac {dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              tg
              ⁡
              c
              x
              −
              1
            
          
        
        =
        −
        
          
            x
            2
          
        
        +
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
        
        sin
        ⁡
        c
        x
        −
        cos
        ⁡
        c
        x
        
          |
        
        
        
      
    
    {\displaystyle \int {\frac {dx}{\operatorname {tg} cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|\,\!}
  
  
    
      
        ∫
        
          
            
              tg
              ⁡
              c
              x
              
              d
              x
            
            
              tg
              ⁡
              c
              x
              +
              1
            
          
        
        =
        
          
            x
            2
          
        
        −
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
        
        sin
        ⁡
        c
        x
        +
        cos
        ⁡
        c
        x
        
          |
        
        
        
      
    
    {\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|\,\!}
  
  
    
      
        ∫
        
          
            
              tg
              ⁡
              c
              x
              
              d
              x
            
            
              tg
              ⁡
              c
              x
              −
              1
            
          
        
        =
        
          
            x
            2
          
        
        +
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
        
        sin
        ⁡
        c
        x
        −
        cos
        ⁡
        c
        x
        
          |
        
        
        
      
    
    {\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|\,\!}
  


== Կոտանգենս ==

  
    
      
        ∫
        cot
        ⁡
        a
        x
        
        
          d
        
        x
        =
        
          
            1
            a
          
        
        ln
        ⁡
        
          |
        
        sin
        ⁡
        a
        x
        
          |
        
        +
        C
        
        
      
    
    {\displaystyle \int \cot ax\;\mathrm {d} x={\frac {1}{a}}\ln |\sin ax|+C\,\!}
  
  
    
      
        ∫
        
          cot
          
            n
          
        
        ⁡
        a
        x
        
        
          d
        
        x
        =
        −
        
          
            1
            
              a
              (
              n
              −
              1
              )
            
          
        
        
          cot
          
            n
            −
            1
          
        
        ⁡
        a
        x
        −
        ∫
        
          cot
          
            n
            −
            2
          
        
        ⁡
        a
        x
        
        
          d
        
        x
        
        
          
            (for 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \cot ^{n}ax\;\mathrm {d} x=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                d
              
              x
            
            
              1
              +
              cot
              ⁡
              a
              x
            
          
        
        =
        ∫
        
          
            
              tan
              ⁡
              a
              x
              
              
                d
              
              x
            
            
              tan
              ⁡
              a
              x
              +
              1
            
          
        
        
        
      
    
    {\displaystyle \int {\frac {\mathrm {d} x}{1+\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax+1}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                d
              
              x
            
            
              1
              −
              cot
              ⁡
              a
              x
            
          
        
        =
        ∫
        
          
            
              tan
              ⁡
              a
              x
              
              
                d
              
              x
            
            
              tan
              ⁡
              a
              x
              −
              1
            
          
        
        
        
      
    
    {\displaystyle \int {\frac {\mathrm {d} x}{1-\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax-1}}\,\!}
  


== Սեկանս ==

  
    
      
        ∫
        sec
        ⁡
        
          a
          x
        
        
        
          d
        
        x
        =
        
          
            1
            a
          
        
        ln
        ⁡
        
          
            |
            
              sec
              ⁡
              
                a
                x
              
              +
              tan
              ⁡
              
                a
                x
              
            
            |
          
        
        +
        C
      
    
    {\displaystyle \int \sec {ax}\,\mathrm {d} x={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}
  
  
    
      
        ∫
        
          sec
          
            2
          
        
        ⁡
        
          x
        
        
        
          d
        
        x
        =
        tan
        ⁡
        
          x
        
        +
        C
      
    
    {\displaystyle \int \sec ^{2}{x}\,\mathrm {d} x=\tan {x}+C}
  
  
    
      
        ∫
        
          sec
          
            3
          
        
        ⁡
        x
        
        d
        x
        =
        
          
            1
            2
          
        
        sec
        ⁡
        x
        tan
        ⁡
        x
        +
        
          
            1
            2
          
        
        ln
        ⁡
        
          |
        
        sec
        ⁡
        x
        +
        tan
        ⁡
        x
        
          |
        
        +
        C
        .
      
    
    {\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C.}
  
  
    
      
        ∫
        
          sec
          
            n
          
        
        ⁡
        
          a
          x
        
        
        
          d
        
        x
        =
        
          
            
              
                sec
                
                  n
                  −
                  2
                
              
              ⁡
              
                a
                x
              
              tan
              ⁡
              
                a
                x
              
            
            
              a
              (
              n
              −
              1
              )
            
          
        
        
        +
        
        
          
            
              n
              −
              2
            
            
              n
              −
              1
            
          
        
        ∫
        
          sec
          
            n
            −
            2
          
        
        ⁡
        
          a
          x
        
        
        
          d
        
        x
        
        
          
             (for 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sec ^{n}{ax}\,\mathrm {d} x={\frac {\sec ^{n-2}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,\mathrm {d} x\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                d
              
              x
            
            
              sec
              ⁡
              
                x
              
              +
              1
            
          
        
        =
        x
        −
        tan
        ⁡
        
          
            x
            2
          
        
        +
        C
      
    
    {\displaystyle \int {\frac {\mathrm {d} x}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}
  


== Կոսեկանս ==

  
    
      
        ∫
        csc
        ⁡
        
          c
          x
        
        
        d
        x
        =
        −
        
          
            1
            c
          
        
        ln
        ⁡
        
          
            |
            
              csc
              ⁡
              
                c
                x
              
              +
              ctg
              ⁡
              
                c
                x
              
            
            |
          
        
      
    
    {\displaystyle \int \csc {cx}\,dx=-{\frac {1}{c}}\ln {\left|\csc {cx}+\operatorname {ctg} {cx}\right|}}
  
  
    
      
        ∫
        
          csc
          
            n
          
        
        ⁡
        
          c
          x
        
        
        d
        x
        =
        −
        
          
            
              
                csc
                
                  n
                  −
                  1
                
              
              ⁡
              
                c
                x
              
              cos
              ⁡
              
                c
                x
              
            
            
              c
              (
              n
              −
              1
              )
            
          
        
        
        +
        
        
          
            
              n
              −
              2
            
            
              n
              −
              1
            
          
        
        ∫
        
          csc
          
            n
            −
            2
          
        
        ⁡
        
          c
          x
        
        
        d
        x
        
        
          
             ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \csc ^{n}{cx}\,dx=-{\frac {\csc ^{n-1}{cx}\cos {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{cx}\,dx\qquad {\mbox{ ( }}n\neq 1{\mbox{)}}\,\!}
  


== Ինտեգրալներ խառը եռանկյունաչափական ֆունկցիաներով ==


=== Սինուս և կոսինուս ===

  
    
      
        ∫
        
          
            
              d
              x
            
            
              cos
              ⁡
              c
              x
              ±
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            
              c
              
                
                  2
                
              
            
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            
              (
              
                
                  
                    
                      c
                      x
                    
                    2
                  
                
                ±
                
                  
                    π
                    8
                  
                
              
              )
            
          
          |
        
      
    
    {\displaystyle \int {\frac {dx}{\cos cx\pm \sin cx}}={\frac {1}{c{\sqrt {2}}}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}\pm {\frac {\pi }{8}}\right)\right|}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              (
              cos
              ⁡
              c
              x
              ±
              sin
              ⁡
              c
              x
              
                )
                
                  2
                
              
            
          
        
        =
        
          
            1
            
              2
              c
            
          
        
        tg
        ⁡
        
          (
          
            c
            x
            ∓
            
              
                π
                4
              
            
          
          )
        
      
    
    {\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}}}={\frac {1}{2c}}\operatorname {tg} \left(cx\mp {\frac {\pi }{4}}\right)}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              (
              cos
              ⁡
              x
              +
              sin
              ⁡
              x
              
                )
                
                  n
                
              
            
          
        
        =
        
          
            1
            
              n
              −
              1
            
          
        
        
          (
          
            
              
                
                  sin
                  ⁡
                  x
                  −
                  cos
                  ⁡
                  x
                
                
                  (
                  cos
                  ⁡
                  x
                  +
                  sin
                  ⁡
                  x
                  
                    )
                    
                      n
                      −
                      1
                    
                  
                
              
            
            −
            2
            (
            n
            −
            2
            )
            ∫
            
              
                
                  d
                  x
                
                
                  (
                  cos
                  ⁡
                  x
                  +
                  sin
                  ⁡
                  x
                  
                    )
                    
                      n
                      −
                      2
                    
                  
                
              
            
          
          )
        
      
    
    {\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}}}={\frac {1}{n-1}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}-2(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2}}}\right)}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
              +
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            x
            2
          
        
        +
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            sin
            ⁡
            c
            x
            +
            cos
            ⁡
            c
            x
          
          |
        
      
    
    {\displaystyle \int {\frac {\cos cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
              −
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            x
            2
          
        
        −
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            sin
            ⁡
            c
            x
            −
            cos
            ⁡
            c
            x
          
          |
        
      
    
    {\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
              +
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            x
            2
          
        
        −
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            sin
            ⁡
            c
            x
            +
            cos
            ⁡
            c
            x
          
          |
        
      
    
    {\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
              −
              sin
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            x
            2
          
        
        −
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            sin
            ⁡
            c
            x
            −
            cos
            ⁡
            c
            x
          
          |
        
      
    
    {\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              c
              x
              
              d
              x
            
            
              sin
              ⁡
              c
              x
              (
              1
              +
              cos
              ⁡
              c
              x
              )
            
          
        
        =
        −
        
          
            1
            
              4
              c
            
          
        
        
          tg
          
            2
          
        
        ⁡
        
          
            
              c
              x
            
            2
          
        
        +
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            
              
                
                  c
                  x
                
                2
              
            
          
          |
        
      
    
    {\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+\cos cx)}}=-{\frac {1}{4c}}\operatorname {tg} ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              c
              x
              
              d
              x
            
            
              sin
              ⁡
              c
              x
              (
              1
              +
              −
              cos
              ⁡
              c
              x
              )
            
          
        
        =
        −
        
          
            1
            
              4
              c
            
          
        
        
          ctg
          
            2
          
        
        ⁡
        
          
            
              c
              x
            
            2
          
        
        −
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            
              
                
                  c
                  x
                
                2
              
            
          
          |
        
      
    
    {\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+-\cos cx)}}=-{\frac {1}{4c}}\operatorname {ctg} ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
              (
              1
              +
              sin
              ⁡
              c
              x
              )
            
          
        
        =
        
          
            1
            
              4
              c
            
          
        
        
          ctg
          
            2
          
        
        ⁡
        
          (
          
            
              
                
                  c
                  x
                
                2
              
            
            +
            
              
                π
                4
              
            
          
          )
        
        +
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            
              (
              
                
                  
                    
                      c
                      x
                    
                    2
                  
                
                +
                
                  
                    π
                    4
                  
                
              
              )
            
          
          |
        
      
    
    {\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1+\sin cx)}}={\frac {1}{4c}}\operatorname {ctg} ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2c}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
              (
              1
              −
              sin
              ⁡
              c
              x
              )
            
          
        
        =
        
          
            1
            
              4
              c
            
          
        
        
          tg
          
            2
          
        
        ⁡
        
          (
          
            
              
                
                  c
                  x
                
                2
              
            
            +
            
              
                π
                4
              
            
          
          )
        
        −
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            
              (
              
                
                  
                    
                      c
                      x
                    
                    2
                  
                
                +
                
                  
                    π
                    4
                  
                
              
              )
            
          
          |
        
      
    
    {\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1-\sin cx)}}={\frac {1}{4c}}\operatorname {tg} ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2c}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}
  
  
    
      
        ∫
        sin
        ⁡
        c
        x
        cos
        ⁡
        c
        x
        
        d
        x
        =
        
          
            1
            
              2
              c
            
          
        
        
          sin
          
            2
          
        
        ⁡
        c
        x
        
        
      
    
    {\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}}\sin ^{2}cx\,\!}
  
  
    
      
        ∫
        sin
        ⁡
        
          c
          
            1
          
        
        x
        cos
        ⁡
        
          c
          
            2
          
        
        x
        
        d
        x
        =
        −
        
          
            
              cos
              ⁡
              (
              
                c
                
                  1
                
              
              +
              
                c
                
                  2
                
              
              )
              x
            
            
              2
              (
              
                c
                
                  1
                
              
              +
              
                c
                
                  2
                
              
              )
            
          
        
        −
        
          
            
              cos
              ⁡
              (
              
                c
                
                  1
                
              
              −
              
                c
                
                  2
                
              
              )
              x
            
            
              2
              (
              
                c
                
                  1
                
              
              −
              
                c
                
                  2
                
              
              )
            
          
        
        
        
          
            ( 
          
        
        
          |
        
        
          c
          
            1
          
        
        
          |
        
        ≠
        
          |
        
        
          c
          
            2
          
        
        
          |
        
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sin c_{1}x\cos c_{2}x\;dx=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          sin
          
            n
          
        
        ⁡
        c
        x
        cos
        ⁡
        c
        x
        
        d
        x
        =
        
          
            1
            
              c
              (
              n
              +
              1
              )
            
          
        
        
          sin
          
            n
            +
            1
          
        
        ⁡
        c
        x
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        sin
        ⁡
        c
        x
        
          cos
          
            n
          
        
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            1
            
              c
              (
              n
              +
              1
              )
            
          
        
        
          cos
          
            n
            +
            1
          
        
        ⁡
        c
        x
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          sin
          
            n
          
        
        ⁡
        c
        x
        
          cos
          
            m
          
        
        ⁡
        c
        x
        
        d
        x
        =
        −
        
          
            
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
              
                cos
                
                  m
                  +
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              n
              +
              m
              )
            
          
        
        +
        
          
            
              n
              −
              1
            
            
              n
              +
              m
            
          
        
        ∫
        
          sin
          
            n
            −
            2
          
        
        ⁡
        c
        x
        
          cos
          
            m
          
        
        ⁡
        c
        x
        
        d
        x
        
        
          
            ( 
          
        
        m
        ,
        n
        >
        0
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx=-{\frac {\sin ^{n-1}cx\cos ^{m+1}cx}{c(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}cx\cos ^{m}cx\;dx\qquad {\mbox{( }}m,n>0{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          sin
          
            n
          
        
        ⁡
        c
        x
        
          cos
          
            m
          
        
        ⁡
        c
        x
        
        d
        x
        =
        
          
            
              
                sin
                
                  n
                  +
                  1
                
              
              ⁡
              c
              x
              
                cos
                
                  m
                  −
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              n
              +
              m
              )
            
          
        
        +
        
          
            
              m
              −
              1
            
            
              n
              +
              m
            
          
        
        ∫
        
          sin
          
            n
          
        
        ⁡
        c
        x
        
          cos
          
            m
            −
            2
          
        
        ⁡
        c
        x
        
        d
        x
        
        
          
            ( 
          
        
        m
        ,
        n
        >
        0
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx={\frac {\sin ^{n+1}cx\cos ^{m-1}cx}{c(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}cx\cos ^{m-2}cx\;dx\qquad {\mbox{( }}m,n>0{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              sin
              ⁡
              c
              x
              cos
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            c
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            c
            x
          
          |
        
      
    
    {\displaystyle \int {\frac {dx}{\sin cx\cos cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} cx\right|}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              sin
              ⁡
              c
              x
              
                cos
                
                  n
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            
              c
              (
              n
              −
              1
              )
              
                cos
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        +
        ∫
        
          
            
              d
              x
            
            
              sin
              ⁡
              c
              x
              
                cos
                
                  n
                  −
                  2
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {dx}{\sin cx\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}+\int {\frac {dx}{\sin cx\cos ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
              cos
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            1
            
              c
              (
              n
              −
              1
              )
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        +
        ∫
        
          
            
              d
              x
            
            
              
                sin
                
                  n
                  −
                  2
                
              
              ⁡
              c
              x
              cos
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {dx}{\sin ^{n}cx\cos cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}+\int {\frac {dx}{\sin ^{n-2}cx\cos cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              sin
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  n
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            
              c
              (
              n
              −
              1
              )
              
                cos
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                sin
                
                  2
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            1
            c
          
        
        sin
        ⁡
        c
        x
        +
        
          
            1
            c
          
        
        ln
        ⁡
        
          |
          
            tg
            ⁡
            
              (
              
                
                  
                    π
                    4
                  
                
                +
                
                  
                    
                      c
                      x
                    
                    2
                  
                
              
              )
            
          
          |
        
      
    
    {\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos cx}}=-{\frac {1}{c}}\sin cx+{\frac {1}{c}}\ln \left|\operatorname {tg} \left({\frac {\pi }{4}}+{\frac {cx}{2}}\right)\right|}
  
  
    
      
        ∫
        
          
            
              
                sin
                
                  2
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  n
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            
              sin
              ⁡
              c
              x
            
            
              c
              (
              n
              −
              1
              )
              
                cos
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        −
        
          
            1
            
              n
              −
              1
            
          
        
        ∫
        
          
            
              d
              x
            
            
              
                cos
                
                  n
                  −
                  2
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              n
              −
              1
              )
            
          
        
        +
        ∫
        
          
            
              
                sin
                
                  n
                  −
                  2
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              cos
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos cx}}=-{\frac {\sin ^{n-1}cx}{c(n-1)}}+\int {\frac {\sin ^{n-2}cx\;dx}{\cos cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            
              
                sin
                
                  n
                  +
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              m
              −
              1
              )
              
                cos
                
                  m
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        −
        
          
            
              n
              −
              m
              +
              2
            
            
              m
              −
              1
            
          
        
        ∫
        
          
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  m
                  −
                  2
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        m
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n+1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              n
              −
              m
              )
              
                cos
                
                  m
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        +
        
          
            
              n
              −
              1
            
            
              n
              −
              m
            
          
        
        ∫
        
          
            
              
                sin
                
                  n
                  −
                  2
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        m
        ≠
        n
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}=-{\frac {\sin ^{n-1}cx}{c(n-m)\cos ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}cx\;dx}{\cos ^{m}cx}}\qquad {\mbox{( }}m\neq n{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              m
              −
              1
              )
              
                cos
                
                  m
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        −
        
          
            
              n
              −
              1
            
            
              m
              −
              1
            
          
        
        ∫
        
          
            
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  m
                  −
                  2
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        m
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n-1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-1}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              cos
              ⁡
              c
              x
              
              d
              x
            
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            1
            
              c
              (
              n
              −
              1
              )
              
                sin
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                cos
                
                  2
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              sin
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            c
          
        
        
          (
          
            cos
            ⁡
            c
            x
            +
            ln
            ⁡
            
              |
              
                tg
                ⁡
                
                  
                    
                      c
                      x
                    
                    2
                  
                
              
              |
            
          
          )
        
      
    
    {\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|\right)}
  
  
    
      
        ∫
        
          
            
              
                cos
                
                  2
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                sin
                
                  n
                
              
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            1
            
              n
              −
              1
            
          
        
        
          (
          
            
              
                
                  cos
                  ⁡
                  c
                  x
                
                
                  c
                  
                    sin
                    
                      n
                      −
                      1
                    
                  
                  ⁡
                  c
                  x
                  )
                
              
            
            +
            ∫
            
              
                
                  d
                  x
                
                
                  
                    sin
                    
                      n
                      −
                      2
                    
                  
                  ⁡
                  c
                  x
                
              
            
          
          )
        
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
      
    
    {\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{n-1}}\left({\frac {\cos cx}{c\sin ^{n-1}cx)}}+\int {\frac {dx}{\sin ^{n-2}cx}}\right)\qquad {\mbox{( }}n\neq 1{\mbox{)}}}
  
  
    
      
        ∫
        
          
            
              
                cos
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                sin
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            
              
                cos
                
                  n
                  +
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              m
              −
              1
              )
              
                sin
                
                  m
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        −
        
          
            
              n
              −
              m
              −
              2
            
            
              m
              −
              1
            
          
        
        ∫
        
          
            
              c
              o
              
                s
                
                  n
                
              
              c
              x
              
              d
              x
            
            
              
                sin
                
                  m
                  −
                  2
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        m
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n+1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-m-2}{m-1}}\int {\frac {cos^{n}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                cos
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                sin
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            
              
                cos
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              n
              −
              m
              )
              
                sin
                
                  m
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        +
        
          
            
              n
              −
              1
            
            
              n
              −
              m
            
          
        
        ∫
        
          
            
              c
              o
              
                s
                
                  n
                  −
                  2
                
              
              c
              x
              
              d
              x
            
            
              
                sin
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        m
        ≠
        n
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}={\frac {\cos ^{n-1}cx}{c(n-m)\sin ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m}cx}}\qquad {\mbox{( }}m\neq n{\mbox{)}}\,\!}
  
  
    
      
        ∫
        
          
            
              
                cos
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                sin
                
                  m
                
              
              ⁡
              c
              x
            
          
        
        =
        −
        
          
            
              
                cos
                
                  n
                  −
                  1
                
              
              ⁡
              c
              x
            
            
              c
              (
              m
              −
              1
              )
              
                sin
                
                  m
                  −
                  1
                
              
              ⁡
              c
              x
            
          
        
        −
        
          
            
              n
              −
              1
            
            
              m
              −
              1
            
          
        
        ∫
        
          
            
              c
              o
              
                s
                
                  n
                  −
                  2
                
              
              c
              x
              
              d
              x
            
            
              
                sin
                
                  m
                  −
                  2
                
              
              ⁡
              c
              x
            
          
        
        
        
          
            ( 
          
        
        m
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n-1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}
  


=== Սինուս և տանգենս ===

  
    
      
        ∫
        sin
        ⁡
        c
        x
        tg
        ⁡
        c
        x
        
        d
        x
        =
        
          
            1
            c
          
        
        (
        ln
        ⁡
        
          |
        
        sec
        ⁡
        c
        x
        +
        tg
        ⁡
        c
        x
        
          |
        
        −
        sin
        ⁡
        c
        x
        )
        
        
      
    
    {\displaystyle \int \sin cx\operatorname {tg} cx\;dx={\frac {1}{c}}(\ln |\sec cx+\operatorname {tg} cx|-\sin cx)\,\!}
  
  
    
      
        ∫
        
          
            
              
                tg
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                sin
                
                  2
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            
              c
              (
              n
              −
              1
              )
            
          
        
        
          tg
          
            n
            −
            1
          
        
        ⁡
        (
        c
        x
        )
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n-1)}}\operatorname {tg} ^{n-1}(cx)\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  


=== Կոսինուս և տանգենս ===

  
    
      
        ∫
        
          
            
              
                tg
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  2
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            
              c
              (
              n
              +
              1
              )
            
          
        
        
          tg
          
            n
            +
            1
          
        
        ⁡
        c
        x
        
        
          
            ( 
          
        
        n
        ≠
        −
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\operatorname {tg} ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}
  


=== Սինուս և կոտանգենս ===

  
    
      
        ∫
        
          
            
              
                ctg
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                sin
                
                  2
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            
              c
              (
              n
              +
              1
              )
            
          
        
        
          ctg
          
            n
            +
            1
          
        
        ⁡
        c
        x
        
        
          
            ( 
          
        
        n
        ≠
        −
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n+1)}}\operatorname {ctg} ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}
  


=== Կոսինուս և կոտանգենս ===

  
    
      
        ∫
        
          
            
              
                ctg
                
                  n
                
              
              ⁡
              c
              x
              
              d
              x
            
            
              
                cos
                
                  2
                
              
              ⁡
              c
              x
            
          
        
        =
        
          
            1
            
              c
              (
              1
              −
              n
              )
            
          
        
        
          tg
          
            1
            −
            n
          
        
        ⁡
        c
        x
        
        
          
            ( 
          
        
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\operatorname {tg} ^{1-n}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}
  


=== Տանգենս և կոտանգենս ===

  
    
      
        ∫
        
          
            
              
                tg
                
                  m
                
              
              ⁡
              (
              c
              x
              )
            
            
              
                ctg
                
                  n
                
              
              ⁡
              (
              c
              x
              )
            
          
        
        
        d
        x
        =
        
          
            1
            
              c
              (
              m
              +
              n
              −
              1
              )
            
          
        
        
          tg
          
            m
            +
            n
            −
            1
          
        
        ⁡
        (
        c
        x
        )
        −
        ∫
        
          
            
              
                tg
                
                  m
                  −
                  2
                
              
              ⁡
              (
              c
              x
              )
            
            
              
                ctg
                
                  n
                
              
              ⁡
              (
              c
              x
              )
            
          
        
        
        d
        x
        
        
          
            ( 
          
        
        m
        +
        n
        ≠
        1
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {\operatorname {tg} ^{m}(cx)}{\operatorname {ctg} ^{n}(cx)}}\;dx={\frac {1}{c(m+n-1)}}\operatorname {tg} ^{m+n-1}(cx)-\int {\frac {\operatorname {tg} ^{m-2}(cx)}{\operatorname {ctg} ^{n}(cx)}}\;dx\qquad {\mbox{( }}m+n\neq 1{\mbox{)}}\,\!}
  


== Տես նաև ==
Ֆունկցիայի ածանցյալ
Ածանցելի ֆունկցիաներ


== Գրականություն ==


=== Գրքեր ===
Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений (4-е издание). М.։ Наука, 1963. ISBN 0-12-294757-6 // EqWorld
Двайт Г. Б. Таблицы интегралов СПб։ «Издательство и типография АО ВНИИГ им. Б. В. Веденеева», 1995.-176 с. ISBN 5-85529-029-8.
D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4


=== Ինտեգրալների աղյուսակներ ===
Ինտեգրալներ EqWorld-ում
S.O.S. Մաթեմատիկա


=== Ինտեգրալների հաշվում ===
The Integrator (Wolfram Research)
Թվեր