Lista de integrais de funcións trigonométricas

A seguinte é unha lista de integrais de funcións trigonométricas. Integrais que conteñen soamente sin
∫ 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 (para 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{(para }}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 n sin ⁡ c x d x
− x n c cos ⁡ c x + n c ∫ x n − 1 cos ⁡ c x d x (para 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{(para }}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 | tan ⁡ c x 2 | {\displaystyle \int {\frac {dx}{\sin cx}}
{\frac {1}{c}}\left|\tan {\frac {cx}{2}}\right|} ∫ d x sin n ⁡ c x
cos ⁡ c x c ( n − 1 ) sin n − 1 ⁡ c x + n − 2 n − 1 ∫ d x sin n − 2 ⁡ c x (para n > 1 ) {\displaystyle \int {\frac {dx}{\sin ^{n}cx}}
{\frac {\cos cx}{c(n-1)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad {\mbox{(para }}n>1{\mbox{)}}} ∫ d x 1 ± sin ⁡ c x
1 c tan ⁡ ( c x 2 ∓ π 4 ) {\displaystyle \int {\frac {dx}{1\pm \sin cx}}
{\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)} ∫ x d x 1 + sin ⁡ c x
x c tan ⁡ ( 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}}\tan \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 cot ⁡ ( 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}}\cot \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 tan ⁡ ( p i 4 ∓ c x 2 ) {\displaystyle \int {\frac {\sin cx\; dx}{1\pm \sin cx}}
\pm x+{\frac {1}{c}}\tan \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 ) (para | 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{(para }}|c_{1}|\neq |c_{2}|{\mbox{)}}}
Integrais que conteñen soamente cos
∫ cos ⁡ c x d x
1 c sin ⁡ c x {\displaystyle \int \cos cx\; dx
{\frac {1}{c}}\sin cx} ∫ cos n ⁡ c x d x
− cos n − 1 ⁡ c x sin ⁡ c x n c + n − 1 n ∫ cos n − 2 ⁡ c x d x (para n > 0 ) {\displaystyle \int \cos ^{n}cx\; dx
-{\frac {\cos ^{n-1}cx\sin cx}{nc}}+{\frac {n-1}{n}}\int \cos ^{n-2}cx\; dx\qquad {\mbox{(para }}n>0{\mbox{)}}} ∫ x cos ⁡ c x d x
cos ⁡ c x c 2 + x sin ⁡ c x c {\displaystyle \int x\cos cx\; dx
{\frac {\cos cx}{c^{2}}}+{\frac {x\sin cx}{c}}} ∫ x n cos ⁡ c x d x
x n sin ⁡ c x c − n c ∫ x n − 1 sin ⁡ c x d x {\displaystyle \int x^{n}\cos cx\; dx
{\frac {x^{n}\sin cx}{c}}-{\frac {n}{c}}\int x^{n-1}\sin cx\; dx} ∫ cos ⁡ c x x d x
ln ⁡ | c x | + ∑ i
1 ∞ ( − 1 ) i ( c x ) 2 i 2 i ⋅ ( 2 i )! {\displaystyle \int {\frac {\cos cx}{x}}dx
\ln |cx|+\sum _{i
1}^{\infty }(-1)^{i}{\frac {(cx)^{2i}}{2i\cdot (2i)! }}} ∫ cos ⁡ c x x n d x
− cos ⁡ c x ( n − 1 ) x n − 1 − c n − 1 ∫ sin ⁡ c x x n − 1 d x (para n ≠ 1 ) {\displaystyle \int {\frac {\cos cx}{x^{n}}}dx
-{\frac {\cos cx}{(n-1)x^{n-1}}}-{\frac {c}{n-1}}\int {\frac {\sin cx}{x^{n-1}}}dx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ d x cos ⁡ c x
1 c ln ⁡ | tan ⁡ ( c x 2 + π 4 ) | {\displaystyle \int {\frac {dx}{\cos cx}}
{\frac {1}{c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|} ∫ d x cos n ⁡ c x
sin ⁡ c x c ( n − 1 ) c o s n − 1 c x + n − 2 n − 1 ∫ d x cos n − 2 ⁡ c x (para n > 1 ) {\displaystyle \int {\frac {dx}{\cos ^{n}cx}}
{\frac {\sin cx}{c(n-1)cos^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{(para }}n>1{\mbox{)}}} ∫ d x 1 + cos ⁡ c x
1 c tan ⁡ c x 2 {\displaystyle \int {\frac {dx}{1+\cos cx}}
{\frac {1}{c}}\tan {\frac {cx}{2}}} ∫ d x 1 − cos ⁡ c x
− 1 c cot ⁡ c x 2 {\displaystyle \int {\frac {dx}{1-\cos cx}}
-{\frac {1}{c}}\cot {\frac {cx}{2}}} ∫ x d x 1 + cos ⁡ c x
x c tan ⁡ c x 2 + 2 c 2 ln ⁡ | cos ⁡ c x 2 | {\displaystyle \int {\frac {x\; dx}{1+\cos cx}}
{\frac {x}{c}}\tan {cx}{2}+{\frac {2}{c^{2}}}\ln \left|\cos {\frac {cx}{2}}\right|} ∫ x d x 1 − cos ⁡ c x
− x x cot ⁡ c x 2 + 2 c 2 ln ⁡ | sin ⁡ c x 2 | {\displaystyle \int {\frac {x\; dx}{1-\cos cx}}
-{\frac {x}{x}}\cot {cx}{2}+{\frac {2}{c^{2}}}\ln \left|\sin {\frac {cx}{2}}\right|} ∫ cos ⁡ c x d x 1 + cos ⁡ c x
x − 1 c tan ⁡ c x 2 {\displaystyle \int {\frac {\cos cx\; dx}{1+\cos cx}}
x-{\frac {1}{c}}\tan {\frac {cx}{2}}} ∫ cos ⁡ c x d x 1 − cos ⁡ c x
− x − 1 c cot ⁡ c x 2 {\displaystyle \int {\frac {\cos cx\; dx}{1-\cos cx}}
-x-{\frac {1}{c}}\cot {\frac {cx}{2}}} ∫ cos ⁡ c 1 x cos ⁡ 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 ) (para | c 1 | ≠ | c 2 | ) {\displaystyle \int \cos c_{1}x\cos 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{(para }}|c_{1}|\neq |c_{2}|{\mbox{)}}}
Integrais que conteñen soamente tan
∫ tan ⁡ c x d x
− 1 c ln ⁡ | cos ⁡ c x | {\displaystyle \int \tan cx\; dx
-{\frac {1}{c}}\ln |\cos cx|} ∫ tan n ⁡ c x d x
1 c ( n − 1 ) tan n − 1 ⁡ c x − ∫ tan n − 2 ⁡ c x d x (para ) n ≠ 1 ) {\displaystyle \int \tan ^{n}cx\; dx
{\frac {1}{c(n-1)}}\tan ^{n-1}cx-\int \tan ^{n-2}cx\; dx\qquad {\mbox{(para )}}n\neq 1{\mbox{)}}} ∫ d x tan ⁡ c x + 1
x 2 + 12 c ln ⁡ | sin ⁡ c x + cos ⁡ c x | {\displaystyle \int {\frac {dx}{\tan cx+1}}
{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|} ∫ d x tan ⁡ c x − 1
− x 2 + 12 c ln ⁡ | sin ⁡ c x − cos ⁡ c x | {\displaystyle \int {\frac {dx}{\tan cx-1}}
-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|} ∫ tan ⁡ c x d x tan ⁡ c x + 1
x 2 − 12 c ln ⁡ | sin ⁡ c x + cos ⁡ c x | {\displaystyle \int {\frac {\tan cx\; dx}{\tan cx+1}}
{\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|} ∫ tan ⁡ c x d x tan ⁡ c x − 1
x 2 + 12 c ln ⁡ | sin ⁡ c x − cos ⁡ c x | {\displaystyle \int {\frac {\tan cx\; dx}{\tan cx-1}}
{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|}
Integrais que conteñen soamente cot
∫ cot ⁡ c x d x
1 c ln ⁡ | sin ⁡ c x | {\displaystyle \int \cot cx\; dx
{\frac {1}{c}}\ln |\sin cx|} ∫ cot n ⁡ c x d x
− 1 c ( n − 1 ) cot n − 1 ⁡ c x − ∫ cot n − 2 ⁡ c x d x (para ) n ≠ 1 ) {\displaystyle \int \cot ^{n}cx\; dx
-{\frac {1}{c(n-1)}}\cot ^{n-1}cx-\int \cot ^{n-2}cx\; dx\qquad {\mbox{(para )}}n\neq 1{\mbox{)}}} ∫ d x 1 + cot ⁡ c x
∫ tan ⁡ c x d x tan ⁡ c x + 1 {\displaystyle \int {\frac {dx}{1+\cot cx}}
\int {\frac {\tan cx\; dx}{\tan cx+1}}} ∫ d x 1 − cot ⁡ c x
∫ tan ⁡ c x d x tan ⁡ c x − 1 {\displaystyle \int {\frac {dx}{1-\cot cx}}
\int {\frac {\tan cx\; dx}{\tan cx-1}}}
Integrais que conteñen sin e cos
∫ d x cos ⁡ c x ± sin ⁡ c x
1 c 2 ln ⁡ | tan ⁡ ( c x 2 ± π 8 ) | {\displaystyle \int {\frac {dx}{\cos cx\pm \sin cx}}
{\frac {1}{c{\sqrt {2}}}}\ln \left|\tan \left({\frac {cx}{2}}\pm {\frac {\pi }{8}}\right)\right|} ∫ d x ( cos ⁡ c x ± sin ⁡ c x ) 2
12 c tan ⁡ ( c x ∓ π 4 ) {\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}}}
{\frac {1}{2c}}\tan \left(cx\mp {\frac {\pi }{4}}\right)} ∫ cos ⁡ c x d x cos ⁡ c x + sin ⁡ c x
x 2 + 12 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 − 12 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 − 12 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 − 12 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 )
− 14 c tan 2 ⁡ c x 2 + 12 c ln ⁡ | tan ⁡ c x 2 | {\displaystyle \int {\frac {\cos cx\; dx}{\sin cx(1+\cos cx)}}
-{\frac {1}{4c}}\tan ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|} ∫ cos ⁡ c x d x sin ⁡ c x ( 1 + − cos ⁡ c x )
− 14 c cot 2 ⁡ c x 2 − 12 c ln ⁡ | tan ⁡ c x 2 | {\displaystyle \int {\frac {\cos cx\; dx}{\sin cx(1+-\cos cx)}}
-{\frac {1}{4c}}\cot ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|} ∫ sin ⁡ c x d x cos ⁡ c x ( 1 + sin ⁡ c x )
14 c cot 2 ⁡ ( c x 2 + π 4 ) + 12 c ln ⁡ | tan ⁡ ( c x 2 + π 4 ) | {\displaystyle \int {\frac {\sin cx\; dx}{\cos cx(1+\sin cx)}}
{\frac {1}{4c}}\cot ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|} ∫ sin ⁡ c x d x cos ⁡ c x ( 1 − sin ⁡ c x )
14 c tan 2 ⁡ ( c x 2 + π 4 ) − 12 c ln ⁡ | tan ⁡ ( c x 2 + π 4 ) | {\displaystyle \int {\frac {\sin cx\; dx}{\cos cx(1-\sin cx)}}
{\frac {1}{4c}}\tan ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|} ∫ sin ⁡ c x cos ⁡ c x d x
12 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 ) (para | 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{(para }}|c_{1}|\neq |c_{2}|{\mbox{)}}} ∫ sin n ⁡ c x cos ⁡ c x d x
1 c ( n + 1 ) sin n + 1 ⁡ c x (para n ≠ 1 ) {\displaystyle \int \sin ^{n}cx\cos cx\; dx
{\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ sin ⁡ c x cos n ⁡ c x d x
− 1 c ( n + 1 ) cos n + 1 ⁡ c x (para n ≠ 1 ) {\displaystyle \int \sin cx\cos ^{n}cx\; dx
-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{(para }}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 (para 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{(para }}m, n>0{\mbox{)}}} tamén: ∫ 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 (para 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{(para }}m, n>0{\mbox{)}}} ∫ d x sin ⁡ c x cos ⁡ c x
1 c ln ⁡ | tan ⁡ c x | {\displaystyle \int {\frac {dx}{\sin cx\cos cx}}
{\frac {1}{c}}\ln \left|\tan 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 (para 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{(para }}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 (para 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{(para }}n\neq 1{\mbox{)}}} ∫ sin ⁡ c x d x cos n ⁡ c x
1 c ( n − 1 ) cos n − 1 ⁡ c x (para n ≠ 1 ) {\displaystyle \int {\frac {\sin cx\; dx}{\cos ^{n}cx}}
{\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ sin 2 ⁡ c x d x cos ⁡ c x
− 1 c sin ⁡ c x + 1 c ln ⁡ | tan ⁡ ( π 4 + c x 2 ) | {\displaystyle \int {\frac {\sin ^{2}cx\; dx}{\cos cx}}
-{\frac {1}{c}}\sin cx+{\frac {1}{c}}\ln \left|\tan \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 (para 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{(para }}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 (for 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{(for }}n\neq 1{\mbox{)}}} ∫ s i n 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 (para 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{(para }}m\neq 1{\mbox{)}}} tamén: ∫ s i n 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 (para 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{(para }}m\neq n{\mbox{)}}} tamén: ∫ s i n n c x d x cos m ⁡ c x
sin n − 1 ⁡ c x c ( m − 1 ) cos m − 1 ⁡ c x − n − 1 n − 1 ∫ sin n − 1 ⁡ c x d x cos m − 2 ⁡ c x (para 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}{n-1}}\int {\frac {\sin ^{n-1}cx\; dx}{\cos ^{m-2}cx}}\qquad {\mbox{(para }}m\neq 1{\mbox{)}}} ∫ cos ⁡ c x d x sin n ⁡ c x
− 1 c ( n − 1 ) sin n − 1 ⁡ c x (para n ≠ 1 ) {\displaystyle \int {\frac {\cos cx\; dx}{\sin ^{n}cx}}
-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{(para }}n\neq 1{\mbox{)}}} ∫ cos 2 ⁡ c x d x sin ⁡ c x
1 c ( cos ⁡ c x + ln ⁡ | tan ⁡ c x 2 | ) {\displaystyle \int {\frac {\cos ^{2}cx\; dx}{\sin cx}}
{\frac {1}{c}}\left(\cos cx+\ln \left|\tan {\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 ) (para 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{(para }}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 (para 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{(para }}m\neq 1{\mbox{)}}} tamén: ∫ 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 (para 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{(para }}m\neq n{\mbox{)}}} tamén: ∫ 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 (para 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{(para }}m\neq 1{\mbox{)}}}
Integrais que conteñen cos e tan
∫ tan n ⁡ c x d x cos 2 ⁡ c x
1 c ( n + 1 ) tan n + 1 ⁡ c x (para n ≠ 1 ) {\displaystyle \int {\frac {\tan ^{n}cx\; dx}{\cos ^{2}cx}}
{\frac {1}{c(n+1)}}\tan ^{n+1}cx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}
Integrais que conteñen sin e cot
∫ cot n ⁡ c x d x s i n 2 c x
1 c ( n + 1 ) cot n + 1 ⁡ c x (para n ≠ 1 ) {\displaystyle \int {\frac {\cot ^{n}cx\; dx}{sin^{2}cx}}
{\frac {1}{c(n+1)}}\cot ^{n+1}cx\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}
Integrais que conteñen cos e cot
Integrais que conteñen tan e cot