有理函数积分表
以下是部份有理函數的積分表。

  
    
      
        ∫
        (
        a
        x
        +
        b
        
          )
          
            n
          
        
        d
        x
        =
        
          
            
              (
              a
              x
              +
              b
              
                )
                
                  n
                  +
                  1
                
              
            
            
              a
              (
              n
              +
              1
              )
            
          
        
        +
        C
        
        (
        n
        ≠
        −
        1
        )
      
    
    {\displaystyle \int (ax+b)^{n} dx={\frac {(ax+b)^{n+1}}{a (n+1)}}+C\qquad (n\neq -1)}
  
  
    
      
        ∫
        
          
            1
            
              a
              x
              +
              b
            
          
        
        d
        x
        =
        
          
            1
            a
          
        
        ln
        ⁡
        
          |
          
            a
            x
            +
            b
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {1}{ax+b}} dx={\frac {1}{a}}\ln \left|ax+b\right|+C}
  
  
    
      
        ∫
        x
        (
        a
        x
        +
        b
        
          )
          
            n
          
        
        d
        x
        =
        
          
            
              a
              (
              n
              +
              1
              )
              x
              −
              b
            
            
              
                a
                
                  2
                
              
              (
              n
              +
              1
              )
              (
              n
              +
              2
              )
            
          
        
        (
        a
        x
        +
        b
        
          )
          
            n
            +
            1
          
        
        +
        C
        
        (
        n
        ∉
        {
        1
        ,
        2
        }
        )
      
    
    {\displaystyle \int x (ax+b)^{n} dx={\frac {a (n+1) x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}+C\qquad (n\not \in \{1,2\})}
  
  
    
      
        ∫
        
          
            x
            
              a
              x
              +
              b
            
          
        
        d
        x
        =
        
          
            x
            a
          
        
        −
        
          
            b
            
              a
              
                2
              
            
          
        
        ln
        ⁡
        
          |
          
            a
            x
            +
            b
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {x}{ax+b}} dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|+C}
  
  
    
      
        ∫
        
          
            x
            
              (
              a
              x
              +
              b
              
                )
                
                  2
                
              
            
          
        
        d
        x
        =
        
          
            b
            
              
                a
                
                  2
                
              
              (
              a
              x
              +
              b
              )
            
          
        
        +
        
          
            1
            
              a
              
                2
              
            
          
        
        ln
        ⁡
        
          |
          
            a
            x
            +
            b
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {x}{(ax+b)^{2}}} dx={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|+C}
  
  
    
      
        ∫
        
          
            x
            
              (
              a
              x
              +
              b
              
                )
                
                  n
                
              
            
          
        
        d
        x
        =
        
          
            
              a
              (
              1
              −
              n
              )
              x
              −
              b
            
            
              
                a
                
                  2
                
              
              (
              n
              −
              1
              )
              (
              n
              −
              2
              )
              (
              a
              x
              +
              b
              
                )
                
                  n
                  −
                  1
                
              
            
          
        
        +
        C
        
        (
        n
        ∉
        {
        1
        ,
        2
        }
        )
      
    
    {\displaystyle \int {\frac {x}{(ax+b)^{n}}} dx={\frac {a (1-n) x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}+C\qquad (n\not \in \{1,2\})}
  
  
    
      
        ∫
        
          
            
              x
              
                2
              
            
            
              a
              x
              +
              b
            
          
        
        d
        x
        =
        
          
            1
            
              a
              
                3
              
            
          
        
        
          [
          
            
              
                
                  (
                  a
                  x
                  +
                  b
                  
                    )
                    
                      2
                    
                  
                
                2
              
            
            −
            2
            b
            (
            a
            x
            +
            b
            )
            +
            
              b
              
                2
              
            
            ln
            ⁡
            
              |
              
                a
                x
                +
                b
              
              |
            
          
          ]
        
        +
        C
      
    
    {\displaystyle \int {\frac {x^{2}}{ax+b}} dx={\frac {1}{a^{3}}}\left [{\frac {(ax+b)^{2}}{2}}-2b (ax+b)+b^{2}\ln \left|ax+b\right|\right]+C}
  
  
    
      
        ∫
        
          
            
              x
              
                2
              
            
            
              (
              a
              x
              +
              b
              
                )
                
                  2
                
              
            
          
        
        d
        x
        =
        
          
            1
            
              a
              
                3
              
            
          
        
        
          (
          
            a
            x
            +
            b
            −
            2
            b
            ln
            ⁡
            
              |
              
                a
                x
                +
                b
              
              |
            
            −
            
              
                
                  b
                  
                    2
                  
                
                
                  a
                  x
                  +
                  b
                
              
            
          
          )
        
        +
        C
      
    
    {\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}} dx={\frac {1}{a^{3}}}\left (ax+b-2b\ln \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)+C}
  
  
    
      
        ∫
        
          
            
              x
              
                2
              
            
            
              (
              a
              x
              +
              b
              
                )
                
                  3
                
              
            
          
        
        d
        x
        =
        
          
            1
            
              a
              
                3
              
            
          
        
        
          [
          
            ln
            ⁡
            
              |
              
                a
                x
                +
                b
              
              |
            
            +
            
              
                
                  2
                  b
                
                
                  a
                  x
                  +
                  b
                
              
            
            −
            
              
                
                  b
                  
                    2
                  
                
                
                  2
                  (
                  a
                  x
                  +
                  b
                  
                    )
                    
                      2
                    
                  
                
              
            
          
          ]
        
        +
        C
      
    
    {\displaystyle \int {\frac {x^{2}}{(ax+b)^{3}}} dx={\frac {1}{a^{3}}}\left [\ln \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2 (ax+b)^{2}}}\right]+C}
  
  
    
      
        ∫
        
          
            
              x
              
                2
              
            
            
              (
              a
              x
              +
              b
              
                )
                
                  n
                
              
            
          
        
        d
        x
        =
        
          
            1
            
              a
              
                3
              
            
          
        
        
          [
          
            −
            
              
                1
                
                  (
                  n
                  −
                  3
                  )
                  (
                  a
                  x
                  +
                  b
                  
                    )
                    
                      n
                      −
                      3
                    
                  
                
              
            
            +
            
              
                
                  2
                  b
                
                
                  (
                  n
                  −
                  2
                  )
                  (
                  a
                  x
                  +
                  b
                  
                    )
                    
                      n
                      −
                      2
                    
                  
                
              
            
            −
            
              
                
                  b
                  
                    2
                  
                
                
                  (
                  n
                  −
                  1
                  )
                  (
                  a
                  x
                  +
                  b
                  
                    )
                    
                      n
                      −
                      1
                    
                  
                
              
            
          
          ]
        
        +
        C
        
      
    
    {\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}} dx={\frac {1}{a^{3}}}\left [-{\frac {1}{(n-3)(ax+b)^{n-3}}}+{\frac {2b}{(n-2)(ax+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right]+C\qquad }
  

  
    
      
        (
        n
        ∉
        {
        1
        ,
        2
        ,
        3
        }
        )
      
    
    {\displaystyle (n\not \in \{1,2,3\})}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              x
              (
              a
              x
              +
              b
              )
            
          
        
        =
        −
        
          
            1
            b
          
        
        ln
        ⁡
        
          |
          
            
              
                a
                x
                +
                b
              
              x
            
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {dx}{x (ax+b)}}=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|+C}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              
                x
                
                  2
                
              
              (
              a
              x
              +
              b
              )
            
          
        
        =
        −
        
          
            1
            
              b
              x
            
          
        
        +
        
          
            a
            
              b
              
                2
              
            
          
        
        ln
        ⁡
        
          |
          
            
              
                a
                x
                +
                b
              
              x
            
          
          |
        
        +
        C
      
    
    {\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|+C}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              
                x
                
                  2
                
              
              (
              a
              x
              +
              b
              
                )
                
                  2
                
              
            
          
        
        =
        −
        a
        
          [
          
            
              
                1
                
                  
                    b
                    
                      2
                    
                  
                  (
                  a
                  x
                  +
                  b
                  )
                
              
            
            +
            
              
                1
                
                  a
                  
                    b
                    
                      2
                    
                  
                  x
                
              
            
            −
            
              
                2
                
                  b
                  
                    3
                  
                
              
            
            ln
            ⁡
            
              |
              
                
                  
                    a
                    x
                    +
                    b
                  
                  x
                
              
              |
            
          
          ]
        
        +
        C
      
    
    {\displaystyle \int {\frac {dx}{x^{2}(ax+b)^{2}}}=-a\left [{\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2} x}}-{\frac {2}{b^{3}}}\ln \left|{\frac {ax+b}{x}}\right|\right]+C}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              
                x
                
                  2
                
              
              +
              
                a
                
                  2
                
              
            
          
        
        =
        
          
            1
            a
          
        
        arctan
        ⁡
        
          
            x
            a
          
        
        +
        C
      
    
    {\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              
                x
                
                  2
                
              
              −
              
                a
                
                  2
                
              
            
          
        
        =
        −
        
          
            1
            a
          
        
        
        
          a
          r
          c
          t
          a
          n
          h
        
        
          
            x
            a
          
        
        =
        
          
            1
            
              2
              a
            
          
        
        ln
        ⁡
        
          
            
              a
              −
              x
            
            
              a
              +
              x
            
          
        
        +
        C
        
        
          
            (
          
        
        
          |
        
        x
        
          |
        
        <
        
          |
        
        a
        
          |
        
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}+C\qquad {\mbox {(}}|x|<|a|{\mbox {)}}\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              
                x
                
                  2
                
              
              −
              
                a
                
                  2
                
              
            
          
        
        =
        −
        
          
            1
            a
          
        
        
        
          a
          r
          c
          c
          o
          t
          h
        
        
          
            x
            a
          
        
        =
        
          
            1
            
              2
              a
            
          
        
        ln
        ⁡
        
          
            
              x
              −
              a
            
            
              x
              +
              a
            
          
        
        +
        C
        
        
          
            (
          
        
        
          |
        
        x
        
          |
        
        >
        
          |
        
        a
        
          |
        
        
          
            )
          
        
        
        
      
    
    {\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}+C\qquad {\mbox {(}}|x|>|a|{\mbox {)}}\,\!}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        =
        
          
            2
            
              4
              a
              c
              −
              
                b
                
                  2
                
              
            
          
        
        arctan
        ⁡
        
          
            
              2
              a
              x
              +
              b
            
            
              4
              a
              c
              −
              
                b
                
                  2
                
              
            
          
        
        +
        C
        ,
        (
        4
        a
        c
        −
        
          b
          
            2
          
        
        >
        0
        )
      
    
    {\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+C,(4ac-b^{2}>0)}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        =
        
          
            2
            
              
                b
                
                  2
                
              
              −
              4
              a
              c
            
          
        
        
        
          a
          r
          t
          a
          n
          h
        
        
          
            
              2
              a
              x
              +
              b
            
            
              
                b
                
                  2
                
              
              −
              4
              a
              c
            
          
        
        +
        C
        =
        
          
            1
            
              
                b
                
                  2
                
              
              −
              4
              a
              c
            
          
        
        ln
        ⁡
        
          |
          
            
              
                2
                a
                x
                +
                b
                −
                
                  
                    
                      b
                      
                        2
                      
                    
                    −
                    4
                    a
                    c
                  
                
              
              
                2
                a
                x
                +
                b
                +
                
                  
                    
                      b
                      
                        2
                      
                    
                    −
                    4
                    a
                    c
                  
                
              
            
          
          |
        
        +
        C
        
        
          
            (
          
        
        4
        a
        c
        −
        
          b
          
            2
          
        
        <0)
          
        
      
    
    {\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+C={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|+C\qquad {\mbox {(}} 4ac-b^{2}<0 {\mbox {)}}}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        =
        −
        
          
            2
            
              2
              a
              x
              +
              b
            
          
        
        +
        C
        
        
          
            (
          
        
        4
        a
        c
        −
        
          b
          
            2
          
        
        =
        0
        
          
            )
          
        
      
    
    {\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}+C\qquad {\mbox {(}} 4ac-b^{2}=0 {\mbox {)}}}
  
  
    
      
        ∫
        
          
            x
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        d
        x
        =
        
          
            1
            
              2
              a
            
          
        
        ln
        ⁡
        
          |
          
            a
            
              x
              
                2
              
            
            +
            b
            x
            +
            c
          
          |
        
        −
        
          
            b
            
              2
              a
            
          
        
        ∫
        
          
            
              d
              x
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        +
        C
      
    
    {\displaystyle \int {\frac {x}{ax^{2}+bx+c}} dx={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {dx}{ax^{2}+bx+c}}+C}
  
  
    
      
        ∫
        
          
            
              m
              x
              +
              n
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        d
        x
        =
        
          
            m
            
              2
              a
            
          
        
        ln
        ⁡
        
          |
          
            a
            
              x
              
                2
              
            
            +
            b
            x
            +
            c
          
          |
        
        +
        
          
            
              2
              a
              n
              −
              b
              m
            
            
              a
              
                
                  4
                  a
                  c
                  −
                  
                    b
                    
                      2
                    
                  
                
              
            
          
        
        arctan
        ⁡
        
          
            
              2
              a
              x
              +
              b
            
            
              4
              a
              c
              −
              
                b
                
                  2
                
              
            
          
        
        +
        C
        
        
          
            (
          
        
        4
        a
        c
        −
        
          b
          
            2
          
        
        >
        0
        
          
            )
          
        
      
    
    {\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}} dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a {\sqrt {4ac-b^{2}}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}+C\qquad {\mbox {(}} 4ac-b^{2}>0 {\mbox {)}}}
  
  
    
      
        ∫
        
          
            
              m
              x
              +
              n
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        d
        x
        =
        
          
            m
            
              2
              a
            
          
        
        ln
        ⁡
        
          |
          
            a
            
              x
              
                2
              
            
            +
            b
            x
            +
            c
          
          |
        
        +
        
          
            
              2
              a
              n
              −
              b
              m
            
            
              a
              
                
                  
                    b
                    
                      2
                    
                  
                  −
                  4
                  a
                  c
                
              
            
          
        
        
        
          a
          r
          t
          a
          n
          h
        
        
          
            
              2
              a
              x
              +
              b
            
            
              
                b
                
                  2
                
              
              −
              4
              a
              c
            
          
        
        +
        C
        
        
          
            (
          
        
        4
        a
        c
        −
        
          b
          
            2
          
        
        <0)
          
        
      
    
    {\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}} dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a {\sqrt {b^{2}-4ac}}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+C\qquad {\mbox {(}} 4ac-b^{2}<0 {\mbox {)}}}
  
  
    
      
        ∫
        
          
            
              m
              x
              +
              n
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        d
        x
        =
        
          
            m
            
              2
              a
            
          
        
        ln
        ⁡
        
          |
          
            a
            
              x
              
                2
              
            
            +
            b
            x
            +
            c
          
          |
        
        −
        
          
            
              2
              a
              n
              −
              b
              m
            
            
              a
              (
              2
              a
              x
              +
              b
              )
            
          
        
        +
        C
        
        
          
            (
          
        
        4
        a
        c
        −
        
          b
          
            2
          
        
        =
        0
        
          
            )
          
        
      
    
    {\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}} dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a (2ax+b)}}+C\qquad {\mbox {(}} 4ac-b^{2}=0 {\mbox {)}}}
  
  
    
      
        ∫
        
          
            
              d
              x
            
            
              (
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
              
                )
                
                  n
                
              
            
          
        
        =
        
          
            
              2
              a
              x
              +
              b
            
            
              (
              n
              −
              1
              )
              (
              4
              a
              c
              −
              
                b
                
                  2
                
              
              )
              (
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
              
                )
                
                  n
                  −
                  1
                
              
            
          
        
        +
        
          
            
              (
              2
              n
              −
              3
              )
              2
              a
            
            
              (
              n
              −
              1
              )
              (
              4
              a
              c
              −
              
                b
                
                  2
                
              
              )
            
          
        
        ∫
        
          
            
              d
              x
            
            
              (
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
              
                )
                
                  n
                  −
                  1
                
              
            
          
        
        +
        C
        
        
      
    
    {\displaystyle \int {\frac {dx}{(ax^{2}+bx+c)^{n}}}={\frac {2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}+{\frac {(2n-3) 2a}{(n-1)(4ac-b^{2})}}\int {\frac {dx}{(ax^{2}+bx+c)^{n-1}}}+C\,\!}
  
  
    
      
        ∫
        
          
            x
            
              (
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
              
                )
                
                  n
                
              
            
          
        
        d
        x
        =
        
          
            
              b
              x
              +
              2
              c
            
            
              (
              n
              −
              1
              )
              (
              4
              a
              c
              −
              
                b
                
                  2
                
              
              )
              (
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
              
                )
                
                  n
                  −
                  1
                
              
            
          
        
        −
        
          
            
              b
              (
              2
              n
              −
              3
              )
            
            
              (
              n
              −
              1
              )
              (
              4
              a
              c
              −
              
                b
                
                  2
                
              
              )
            
          
        
        ∫
        
          
            
              d
              x
            
            
              (
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
              
                )
                
                  n
                  −
                  1
                
              
            
          
        
        +
        C
        
        
      
    
    {\displaystyle \int {\frac {x}{(ax^{2}+bx+c)^{n}}} dx={\frac {bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}-{\frac {b (2n-3)}{(n-1)(4ac-b^{2})}}\int {\frac {dx}{(ax^{2}+bx+c)^{n-1}}}+C\,\!}
  對於任意的有理函數，我們都能通過部分分式 (partial fraction) 把該函數分拆為數個函數的總和，其中每個函數符合以下的形式：
  
    
      
        
          
            
              p
              x
              +
              q
            
            
              
                (
                
                  a
                  
                    x
                    
                      2
                    
                  
                  +
                  b
                  x
                  +
                  c
                
                )
              
              
                n
              
            
          
        
      
    
    {\displaystyle {\frac {px+q}{\left (ax^{2}+bx+c\right)^{n}}}}
  。我們繼而能把每一個該種形式的函數作積分運算：

  
    
      
        ∫
        
          
            
              d
              x
            
            
              x
              (
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
              )
            
          
        
        =
        
          
            1
            
              2
              c
            
          
        
        ln
        ⁡
        
          |
          
            
              
                x
                
                  2
                
              
              
                a
                
                  x
                  
                    2
                  
                
                +
                b
                x
                +
                c
              
            
          
          |
        
        −
        
          
            b
            
              2
              c
            
          
        
        ∫
        
          
            
              d
              x
            
            
              a
              
                x
                
                  2
                
              
              +
              b
              x
              +
              c
            
          
        
        +
        C
      
    
    {\displaystyle \int {\frac {dx}{x (ax^{2}+bx+c)}}={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {dx}{ax^{2}+bx+c}}+C}