Origem: Wikipédia, a enciclopédia livre.
A lista seguinte contém integrais de funções trigonométricas.
A constante "c" é assumida como não nula.
Integrais de funções trigonométricas contendo apenas seno[editar | editar código-fonte]
![{\displaystyle \int \operatorname {sen} cx\;dx=-{\frac {1}{c}}\cos cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c02c172967ba86a4c234d991298364e08fe817a)
![{\displaystyle \int \operatorname {sen} ^{n}cx\;dx=-{\frac {\operatorname {sen} ^{n-1}cx\cos cx}{nc}}+{\frac {n-1}{n}}\int \operatorname {sen} ^{n-2}cx\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84980ed9de2c85535a39ec59af436fe37bc9423e)
![{\displaystyle \int {\sqrt {1-\operatorname {sen} {x}}}\,dx=\int {\sqrt {\operatorname {cvs} {x}}}\,dx=2{\frac {\cos {\frac {x}{2}}+\operatorname {sen} {\frac {x}{2}}}{\cos {\frac {x}{2}}-\operatorname {sen} {\frac {x}{2}}}}{\sqrt {\operatorname {cvs} {x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb2193e756727699e337cc5a27a6e228643c803)
onde cvs{x} é a função de Coversene
![{\displaystyle \int x\operatorname {sen} cx\;dx={\frac {\operatorname {sen} cx}{c^{2}}}-{\frac {x\cos cx}{c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6672602c8c7c189e3b036166d9ff839f3c2c21df)
![{\displaystyle \int x^{n}\operatorname {sen} cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa614cab44645e1a3dc6cfcd509d46435b78162d)
![{\displaystyle \int {\frac {\operatorname {sen} cx}{x}}dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2bf83a21eb491873fb539f25a158b509dbbd66)
![{\displaystyle \int {\frac {\operatorname {sen} cx}{x^{n}}}dx=-{\frac {\sin cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos cx}{x^{n-1}}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5305f2235389c19c497f5be06f8bf29169c3158e)
![{\displaystyle \int {\frac {dx}{\operatorname {sen} cx}}={\frac {1}{c}}\ln \left|\tan {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/931f5d6694177a34cf857de61900191d8f9797a3)
![{\displaystyle \int {\frac {dx}{\operatorname {sen} ^{n}cx}}={\frac {\cos cx}{c(1-n)\operatorname {sen} ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\operatorname {sen} ^{n-2}cx}}\qquad {\mbox{(for }}n>1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb49c1cd6e95485b9728b3caf0fe40332709ae1f)
![{\displaystyle \int {\frac {dx}{1\pm \operatorname {sen} cx}}={\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/143796623c2415718b4d42202fbf66f54a74c400)
![{\displaystyle \int {\frac {x\;dx}{1+\operatorname {sen} 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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c644dc1b9a3e3cdcc44b6f4f4b4ae7eb1f33e126)
![{\displaystyle \int {\frac {x\;dx}{1-\operatorname {sen} cx}}={\frac {x}{c}}\cot \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\operatorname {sen} \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afca20d35b55e777a01c42af73544f8d241c3acd)
![{\displaystyle \int {\frac {\operatorname {sen} cx\;dx}{1\pm \operatorname {sen} cx}}=\pm x+{\frac {1}{c}}\tan \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b47fb3a5d82bb9fe6683a5a52674db8d7479d43)
![{\displaystyle \int \operatorname {sen} c_{1}x\operatorname {sen} c_{2}x\;dx={\frac {\operatorname {sen}(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\operatorname {sen}(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(for }}|c_{1}|\neq |c_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/178f207c4cdf4cc50f20c209955ee506a40c1e8c)
![{\displaystyle \int \cos cx\;dx={\frac {1}{c}}\operatorname {sen} cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfce3a23c86baf9dfd99e923232643ef3ed85960)
![{\displaystyle \int \cos ^{n}cx\;dx={\frac {\cos ^{n-1}cx\operatorname {sen} cx}{nc}}+{\frac {n-1}{n}}\int \cos ^{n-2}cx\;dx\qquad {\mbox{(para }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35f794433407d44196f597d0a164f606a5c5a2d7)
![{\displaystyle \int x\cos cx\;dx={\frac {\cos cx}{c^{2}}}+{\frac {x\operatorname {sen} cx}{c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7926167e650886892489391fb33b0dfdd1cf577d)
CALC
![{\displaystyle \int x^{n}\cos cx\;dx={\frac {x^{n}\sin cx}{c}}-{\frac {n}{c}}\int x^{n-1}\sin cx\;dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69bd19d833e7203082060d3c542dde5a0ff8c1d)
![{\displaystyle \int {\frac {\cos cx}{x}}dx=\ln |cx|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(cx)^{2i}}{2i\cdot (2i)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ef8f8e1cdb401b716fc13f8e69becc81203092)
![{\displaystyle \int {\frac {\cos cx}{x^{n}}}dx=-{\frac {\cos cx}{(n-1)x^{n-1}}}-{\frac {c}{n-1}}\int {\frac {\operatorname {sen} cx}{x^{n-1}}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee36b380624f0cd2f2935523540f4628db743d22)
![{\displaystyle \int {\frac {dx}{\cos cx}}={\frac {1}{c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/136d5e28887252d271d77638ae7c15b2b32c09f6)
![{\displaystyle \int {\frac {dx}{\cos ^{n}cx}}={\frac {\operatorname {sen} cx}{c(n-1)cos^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{(for }}n>1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d02c5f12664cac3e2ab25fc0bb91e51a4b96eab0)
![{\displaystyle \int {\frac {dx}{1+\cos cx}}={\frac {1}{c}}\tan {\frac {cx}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3857e4a4dc5840a94aaf5699a09777ad595f175b)
![{\displaystyle \int {\frac {dx}{1-\cos cx}}=-{\frac {1}{c}}\cot {\frac {cx}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29d128d60dc3656d5932ac493b31c88c1fccd275)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8bc9c52bb2adf9e3f5a1392eb83cf837d76134)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef96f5190363e604829cdb7846fc42c786a063b8)
![{\displaystyle \int {\frac {\cos cx\;dx}{1+\cos cx}}=x-{\frac {1}{c}}\tan {\frac {cx}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a8ea6adef5f8c9744dec5adff38976624a609e)
![{\displaystyle \int {\frac {\cos cx\;dx}{1-\cos cx}}=-x-{\frac {1}{c}}\cot {\frac {cx}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/349e26174cbe0dab64b190f8665f39edbc3d2f22)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb08625bd0f7dff32b6a65cdd6bed2229e1e20ee)
![{\displaystyle \int \tan cx\;dx=-{\frac {1}{c}}\ln |\cos cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c470ae53ee37dade58a57ed1afd1f656cad2649c)
![{\displaystyle \int \tan ^{n}cx\;dx={\frac {1}{c(n-1)}}\tan ^{n-1}cx-\int \tan ^{n-2}cx\;dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3976f1b534c4268b9c079e4f89a341c6ee0087f)
![{\displaystyle \int {\frac {dx}{\tan cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55110a30c242159cbf77cc696689c83ee8157b66)
![{\displaystyle \int {\frac {dx}{\tan cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4f0445fdbeeaa31f552a586f0766d1bdd8e37d)
![{\displaystyle \int {\frac {\tan cx\;dx}{\tan cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b584c8e16180f78056b8021b2cca26d303389470)
![{\displaystyle \int {\frac {\tan cx\;dx}{\tan cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceda26c9b1ae3ad8f08de10e40198c168fd9a0bf)
![{\displaystyle \int \sec {cx}\,dx={\frac {1}{c}}\ln {\left|\sec {cx}+\tan {cx}\right|}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/588308331007a93415956219ab2d80eba1749f82)
![{\displaystyle \int \sec ^{n}{cx}\,dx={\frac {\sec ^{n-1}{cx}\sin {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{cx}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed8e933379d51939c937f4f741663ff547bcea6)
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcc12662b9bdd3e263861867ebee100e8d244d7)
![{\displaystyle \int \csc {cx}\,dx=-{\frac {1}{c}}\ln {\left|\csc {cx}-\cot {cx}\right|}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16093acaf7344e6afdb9eb1bfdb002099bd3e9dd)
![{\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{ (for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/239ed6b1ff2641e2d2395047dc5ba7fa7f360c74)
![{\displaystyle \int \cot cx\;dx={\frac {1}{c}}\ln |\sin cx|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68b2ca0e80b80cd16f5d79772ef415340b572b37)
![{\displaystyle \int \cot ^{n}cx\;dx=-{\frac {1}{c(n-1)}}\cot ^{n-1}cx-\int \cot ^{n-2}cx\;dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50852983f2d9657186c7b3f9ce3019ec844eae9b)
![{\displaystyle \int {\frac {dx}{1+\cot cx}}=\int {\frac {\tan cx\;dx}{\tan cx+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69150be9b596c560d9ec1cc9e8e65c0838a3e20d)
![{\displaystyle \int {\frac {dx}{1-\cot cx}}=\int {\frac {\tan cx\;dx}{\tan cx-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3940073b443e32697228788f5d7b7742c94855e8)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddec8e2fcfa6386ad1690c5b1838e9aca4a9c4df)
![{\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}}}={\frac {1}{2c}}\tan \left(cx\mp {\frac {\pi }{4}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af03e2ed5b6774576d1b3cd202dd1e35319fa06)
![{\displaystyle \int {\frac {dx}{(\cos x+\operatorname {sen} x)^{n}}}={\frac {1}{n-1}}\left({\frac {\operatorname {sen} x-\cos x}{(\cos x+\operatorname {sen} x)^{n-1}}}-2(n-2)\int {\frac {dx}{(\cos x+\operatorname {sen} x)^{n-2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d64d88b0e94b895aa5f60dec75ee9bb274efa3ac)
![{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx+\operatorname {sen} cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\operatorname {sen} cx+\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65665722a97fbd83c87bb14329ff56e56f810085)
![{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\operatorname {sen} cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\operatorname {sen} cx-\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c0387480374510d953fe852f156c8ae8674dfa8)
![{\displaystyle \int {\frac {\operatorname {sen} cx\;dx}{\cos cx+\operatorname {sen} cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\operatorname {sen} cx+\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a398fcb0698168ee9609d3aad8fa00742d460c14)
![{\displaystyle \int {\frac {\operatorname {sen} cx\;dx}{\cos cx-\operatorname {sen} cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\operatorname {sen} cx-\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2946117f7c79f159a8629229e2d10002275c7b)
![{\displaystyle \int {\frac {\cos cx\;dx}{\operatorname {sen} cx(1+\cos cx)}}=-{\frac {1}{4c}}\tan ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c89cc959317554a7588fe03d110dd72a3fa68f2f)
![{\displaystyle \int {\frac {\cos cx\;dx}{\operatorname {sen} cx(1+-\cos cx)}}=-{\frac {1}{4c}}\cot ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f64545fde7eae601c4c9ecb445c46b24b4860b1d)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3f98d01f0d576d4ce48b5343e32e79b2327d05)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7103e74e9574f047e5ed9287d0bc3a3d44d79305)
![{\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}}\sin ^{2}cx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99e3ba906a0d014f4539c0267d855cb7aa288d56)
![{\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{(for }}|c_{1}|\neq |c_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6601f43ab80f6118dcb4180c5a17fce3af5b3ac)
![{\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7cd012c2fc3728da6569c9f98bd2f0b812ed73)
![{\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f840b255bae83e246dc7fdd5ef0fada6cff732fd)
![{\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{(for }}m,n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05452ddff05180418e747b51b53f99231d50c16f)
- também:
![{\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{(for }}m,n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc52f5d087dc80b9189f4b5b44a1f8d2f187355a)
![{\displaystyle \int {\frac {dx}{\sin cx\cos cx}}={\frac {1}{c}}\ln \left|\tan cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5c9ba1b9b040eae50c07eb9f47f83561eee42d)
![{\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{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d1a217e6f8cd7ca18bdea02007b50a57197988)
![{\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{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6d48875c6dc496924626842ab6a3dbd1421411)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc558c16c111d12004e597916549e7394f4d789)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7a184b1964664bfc2c1882c1b63b1917c1473f)
![{\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{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7029b668f74c8961df150dbd36d3b8db9f1f77)
![{\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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0122476c0e41344fea1fbacb51420982e39cda)
![{\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{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca10d714b60df2d446d0bbddd7b84c052d9578fa)
- também:
![{\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{(for }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e33afdc98b20255b93b9a088677b4d475f1e015)
- também:
![{\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{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0dc839a427341bec7565ea419a24cd2408eeb03)
![{\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cff4f2d08728cd4b2e3879701747ceb9005e6a8d)
![{\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\tan {\frac {cx}{2}}\right|\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72006b01ffe8fe2d3fb85781b47a33a40c2f7377)
![{\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{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a80f487e490bdb50ee305be4a8baf92a3415804)
![{\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{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3ff75039811ba34b213af414d7ffcf9b4ee103)
- também:
![{\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{(for }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40000288e1161af601c3909a486d92a5621bd897)
- também:
![{\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{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8647b7412fd16fcb0d99fc469a86d72483492df0)
![{\displaystyle \int \operatorname {sen} cx\tan cx\;dx={\frac {1}{c}}(\ln |\sec cx+\tan cx|-\sin cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31c859fbf021bb9d8e55ae6b5dc8182429ac2e51)
![{\displaystyle \int {\frac {\tan ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n-1)}}\tan ^{n-1}(cx)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ebc541f1f14438492896ea7fa5a8f9fee05635)
![{\displaystyle \int {\frac {\tan ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\tan ^{n+1}cx\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dafcd2c188f00e0d6ff0976a56dae776de2c15a)
![{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n+1)}}\cot ^{n+1}cx\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52dc06b3f38fc2fd5d4db8113002f7c9a9d678c3)
![{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\tan ^{1-n}cx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c71ee376d3aae09ba8555b70d21e9e8f506ca69)
![{\displaystyle \int {\frac {\tan ^{m}(cx)}{\cot ^{n}(cx)}}\;dx={\frac {1}{c(m+n-1)}}\tan ^{m+n-1}(cx)-\int {\frac {\tan ^{m-2}(cx)}{\cot ^{n}(cx)}}\;dx\qquad {\mbox{(for }}m+n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89f6a59b1482bac719762db481405031c27d2872)