Tot seguit es presenta una llista de les primitives (o integrals) de funcions trigonomètriques. Per a consultar les integrals que impliquen funcions exponencials i trigonomètriques, veure Llista d'integrals de funcions exponencials. Per a consultar una llista completa de primitives de tota mena de funcions adreceu-vos a taula d'integrals
En totes les fórmules, la constant a se suposa diferent de zero i C indica la constant d'integració.
Integrals de funcions trigonomètriques que inclouen només el sinus
![{\displaystyle \int \sin ax\;dx=-{\frac {1}{a}}\cos ax+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a73eb45066dbbef44abf783f8cac45b6e568f04)
![{\displaystyle \int \sin ^{2}{ax}\;dx={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}-{\frac {1}{2a}}\sin ax\cos ax+C\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/293755ccc64ede76a76570cda68f8bc2eead8f22)
![{\displaystyle \int \sin a_{1}x\sin a_{2}x\;dx={\frac {\sin[(a_{1}-a_{2})x]}{2(a_{1}-a_{2})}}-{\frac {\sin[(a_{1}+a_{2})x]}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(per }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c9d4f206637fabdb93f74961ee73f75fcc47a9)
![{\displaystyle \int \sin ^{n}{ax}\;dx=-{\frac {\sin ^{n-1}ax\cos ax}{na}}+{\frac {n-1}{n}}\int \sin ^{n-2}ax\;dx\qquad {\mbox{(per }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83aa75e476b2e85a30c632d4271b204b91b76987)
![{\displaystyle \int {\frac {dx}{\sin ax}}={\frac {1}{a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/558a7998ef61c2f451a53dc7168710e503e42f8c)
![{\displaystyle \int {\frac {dx}{\sin ^{n}ax}}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}ax}}\qquad {\mbox{(per }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/052badf0b39e0d2d75c368c2a7cb22a310a6d9c6)
![{\displaystyle \int x\sin ax\;dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da65650f139648d3bb59885918799df0c6119f7b)
![{\displaystyle \int x^{n}\sin ax\;dx=-{\frac {x^{n}}{a}}\cos ax+{\frac {n}{a}}\int x^{n-1}\cos ax\;dx\qquad {\mbox{(per }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b178a1e4858572f4e9724d2ad88870dede3c8569)
![{\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(per }}n=2,4,6...{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ea9451e5735fbcc2ee9a43245ee94695fd3453)
![{\displaystyle \int {\frac {\sin ax}{x}}dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe560e92679d92b5099e69ce4be2598e3c2f5ea7)
![{\displaystyle \int {\frac {\sin ax}{x^{n}}}dx=-{\frac {\sin ax}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}\int {\frac {\cos ax}{x^{n-1}}}dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03e7a44d70f97f2ffc9285a1cac5ef28d06f483)
![{\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8439ef42a168ed7e05a7efea83b205790ceb59)
![{\displaystyle \int {\frac {x\;dx}{1+\sin ax}}={\frac {x}{a}}\tan \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{a^{2}}}\ln \left|\cos \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2bd3d7a463e6f185985d416b060c066837a744)
![{\displaystyle \int {\frac {x\;dx}{1-\sin ax}}={\frac {x}{a}}\cot \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)+{\frac {2}{a^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b160118dfa4c5a05cba9a2ebc6526cb1064c9eb)
![{\displaystyle \int {\frac {\sin ax\;dx}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e00c6763dad11fcec444a0db9d7fa534dea659fb)
Integrals de funcions trigonomètriques que inclouen només el cosinus
![{\displaystyle \int \cos ax\;dx={\frac {1}{a}}\sin ax+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23fe2f09db08af7b70bff8e1f52bbc26d4e5e48b)
![{\displaystyle \int \cos ^{n}ax\;dx={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\;dx\qquad {\mbox{(per }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faa92023910b88451b9eaee7f0cbff79eb6c62c1)
![{\displaystyle \int x\cos ax\;dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a224cadde3a4fef31043b5aedd948c765db94c2)
![{\displaystyle \int \cos ^{2}{ax}\;dx={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}+{\frac {1}{2a}}\sin ax\cos ax+C\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e16ba54bc0559f3279fb72e7215f23e4edde65)
![{\displaystyle \int x^{n}\cos ax\;dx={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\;dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d842c2fcaf2158d857e9b67d5da59fca2135497b)
![{\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(per }}n=1,3,5...{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b5092893b20bba81654df6c45056c22c225126)
![{\displaystyle \int {\frac {\cos ax}{x}}dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17e94201a5023bd1897254a8bf7fc4606958637c)
![{\displaystyle \int {\frac {\cos ax}{x^{n}}}dx=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29cb69bc60f150b542767bdd0a40af1f1cc90bc3)
![{\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5013bc2428b1006b40c999d6b427a36f5cf0620)
![{\displaystyle \int {\frac {dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(per }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0273941144e7436f16a9a1dafda160d0cc25b9a9)
![{\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/110e912c95a902109556eaf7954dcde0571fc4c8)
![{\displaystyle \int {\frac {dx}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea932362481541732a94cacdd2885997cdbe7598)
![{\displaystyle \int {\frac {x\;dx}{1+\cos ax}}={\frac {x}{a}}\tan {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\cos {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c75e60562bf09238b854d35ab35bdcbf5c8510)
![{\displaystyle \int {\frac {x\;dx}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5563ed4ebe078ad8616ab89c39b0f832067bdc)
![{\displaystyle \int {\frac {\cos ax\;dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58b792752178d49805802fc55c5a32ff79fc2da0)
![{\displaystyle \int {\frac {\cos ax\;dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1feca0aca8542bf452ecdba3bf778ef0dad786)
![{\displaystyle \int \cos a_{1}x\cos a_{2}x\;dx={\frac {\sin(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+{\frac {\sin(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(per }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b32945765082c7604e31227caa1f6734835720)
Integrals de funcions trigonomètriques que inclouen només la tangent
![{\displaystyle \int \tan ax\;dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec533469c6f4008af58416899e6afccbfed393f)
![{\displaystyle \int \tan ^{n}ax\;dx={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\;dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56d025c61e76d1521439365b8c183ab5a6d9d2e7)
![{\displaystyle \int {\frac {dx}{q\tan ax+p}}={\frac {1}{p^{2}+q^{2}}}(px+{\frac {q}{a}}\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(per }}p^{2}+q^{2}\neq 0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5776e0ded41094e908042f5c9338235f925142f3)
![{\displaystyle \int {\frac {dx}{\tan ax}}={\frac {1}{a}}\ln |\sin ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/865a88c71d447aa57caa031b819e24f3f76a1f38)
![{\displaystyle \int {\frac {dx}{\tan ax+1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1711694feda807b6b6c4d19c1ccda303ef0c02b)
![{\displaystyle \int {\frac {dx}{\tan ax-1}}=-{\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acaa4b1fa8ad7c5675d289c229146ea22ebb1ad)
![{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/401dd109ce2332e369ea8b45ad3e3d8165435b6e)
![{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c3c42a4efe9309b306895d993b1980ddda5e5c)
Integrals de funcions trigonomètriques que inclouen només la secant
![{\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0eae695334d259040d728b565ca374a2c89380)
![{\displaystyle \int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-1}{ax}\sin {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/628446755603c2669d58d2f29507b264389b2479)
[1]
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9acabbd90de19b0d361d572dce3398a57c9d653f)
Integrals de funcions trigonomètriques que inclouen només la cosecant
![{\displaystyle \int \csc {ax}\,dx=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d51921c472f398df353d0db33a9af2ebe5fbf3fb)
![{\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/417803af6cef8535c9b9ee74f75a20ab4180fac0)
![{\displaystyle \int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-1}{ax}\cos {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a96753451f37c1a0fa1aedf4addec5e9d90089d)
Integrals de funcions trigonomètriques que contenen només la cotangent
![{\displaystyle \int \cot ax\;dx={\frac {1}{a}}\ln |\sin ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a390207aa90027cf4e88003ad404e5be9c3ce9)
![{\displaystyle \int \cot ^{n}ax\;dx=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d019451ba74ac82f3a07409a56135510f7dd224)
![{\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb992449b8d41c9ad694ebc79755b1bc6f05b721)
![{\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax-1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae83987f4461f7cefa25e65f6583476d6f6aa1a7)
Integrals de funcions trigonomètriques que inclouen ambdós sinus i cosinus
![{\displaystyle \int {\frac {dx}{\cos ax\pm \sin ax}}={\frac {1}{a{\sqrt {2}}}}\ln \left|\tan \left({\frac {ax}{2}}\pm {\frac {\pi }{8}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f99f9f4158d86f68a6f22ac0b494b8df2a009d24)
![{\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e0e3cebd7eac046797eefb5e8be824a6ec6008)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4a31631ace2155c341f3a42e944454f4d2525b)
![{\displaystyle \int {\frac {\cos ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20dbc04e3f7127782e7b005abd8ee54505f6a3c3)
![{\displaystyle \int {\frac {\cos ax\;dx}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16fc2c648278180416aab94d34a800a261298de)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22bed95877232b1041ff194e4431e3085f8d94bb)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5878e7de15317f1a758a684cd716299d7704f5)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ax(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ceadea9e2db1da77cc7cc229f5dc509b1367de1)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ax(1+-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f23dfa71a079b3aeeab5209cff8bf9b3cfbb33d)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax(1+\sin ax)}}={\frac {1}{4a}}\cot ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b85fee1b43ee4e960660d5506f2fbabee8b8b51f)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax(1-\sin ax)}}={\frac {1}{4a}}\tan ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28889f39f656ab104a2636f836512a46015bc42a)
![{\displaystyle \int \sin ax\cos ax\;dx={\frac {1}{2a}}\sin ^{2}ax+c\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91982650837db8088a6e85410cc1445732288a76)
![{\displaystyle \int \sin a_{1}x\cos a_{2}x\;dx=-{\frac {\cos(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}-{\frac {\cos(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+C\qquad {\mbox{(per }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35bbfca4627754d54099b1e210f3aa9f374797fb)
![{\displaystyle \int \sin ^{n}ax\cos ax\;dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(per }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab151b4e8481a9daf2ed43dac3fdc5ee7b78988)
![{\displaystyle \int \sin ax\cos ^{n}ax\;dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(per }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4525a382cdf606974cd90a41719098434395ec9)
![{\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;dx=-{\frac {\sin ^{n-1}ax\cos ^{m+1}ax}{a(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}ax\cos ^{m}ax\;dx\qquad {\mbox{(per }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76604077a484c710d88fa3e174923a9f8694d265)
- també:
![{\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;dx={\frac {\sin ^{n+1}ax\cos ^{m-1}ax}{a(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}ax\cos ^{m-2}ax\;dx\qquad {\mbox{(per }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/846d2cbeac2f89dc4d10963079bbe2105687445e)
![{\displaystyle \int {\frac {dx}{\sin ax\cos ax}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf404f3f55cfd283adc68644b179ac6bab6f24d7)
![{\displaystyle \int {\frac {dx}{\sin ax\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+\int {\frac {dx}{\sin ax\cos ^{n-2}ax}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfaf0aa6418691ebf353b54f8c607dd09ba6350)
![{\displaystyle \int {\frac {dx}{\sin ^{n}ax\cos ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+\int {\frac {dx}{\sin ^{n-2}ax\cos ax}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/024e0d9fd3cd19c74943573fd23d10b32a748d16)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a68c1370b90d837f58ef1d79bd83ca6f44f5c29)
![{\displaystyle \int {\frac {\sin ^{2}ax\;dx}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/585f6083fc3cb9c10c4ecd369500ce5902de34c6)
![{\displaystyle \int {\frac {\sin ^{2}ax\;dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4502285089f62ef2aec24f3e960f3704f5c201)
![{\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\;dx}{\cos ax}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fda2a532c2fe47417011e37ec4766f9e33ba406)
![{\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}={\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}ax\;dx}{\cos ^{m-2}ax}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b343ab57f6a1d6d0e10dde8c13bda773387d9b8)
- també:
![{\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}=-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}ax\;dx}{\cos ^{m}ax}}\qquad {\mbox{(per }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0826849f53293e26025a688fc7a43f09a6f7cda7)
- també:
![{\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}={\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-2}ax\;dx}{\cos ^{m-2}ax}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4126f154a18d25cd230d98542de32a8ae0da09)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/175625af191190aae815464935a4918eb967242b)
![{\displaystyle \int {\frac {\cos ^{2}ax\;dx}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbc213a769efe305eaa42857879e989b250e33f)
![{\displaystyle \int {\frac {\cos ^{2}ax\;dx}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax)}}+\int {\frac {dx}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02eeb86c1bfccf37633baefef996207c063c9c5e)
![{\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}=-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-m-2}{m-1}}\int {\frac {\cos ^{n}ax\;dx}{\sin ^{m-2}ax}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94429babf0c39f691f939428cf1db79e8fa7ca4e)
- també:
![{\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}={\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cos ^{n-2}ax\;dx}{\sin ^{m}ax}}\qquad {\mbox{(per }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43c781248f0a06e5da95ca1f496b9966c7ad091d)
- també:
![{\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}=-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\cos ^{n-2}ax\;dx}{\sin ^{m-2}ax}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/085ebe0e11a949feb06f233e46893eb889fd4d62)
Integrals de funcions trigonomètriques que inclouen ambdós sinus i tangent
![{\displaystyle \int \sin ax\tan ax\;dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8db0ad9f8d3768112f104d143d00d48a4a10a30)
![{\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c39840e7f7d76566279ee59eed20e9e01cf2d6d)
Integrals de funcions trigonomètriques que inclouen ambdós cosinus i tangent
![{\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(per }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a499a6c8aaff8eca32c0cd35d78aa72cd8684871)
Integrals de funcions trigonomètriques que inclouen ambdós sinus i cotangent
![{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\sin ^{2}ax}}={\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(per }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b64cf9bd89c92265b77f186ee9db59d938162120)
Integrals de funcions trigonomètriques que inclouen ambdós cosinus i cotangent
![{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(per }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/877843b55ba555ce5b293a95a1c0fafb00bcfafe)
Integrals de funcions trigonomètriques amb limits simètrics
![{\displaystyle \int _{-c}^{c}\sin {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33469f374c9c3af903d9607671dcfdf18c1a5077)
![{\displaystyle \int _{-c}^{c}\cos {x}\;dx=2\int _{0}^{c}\cos {x}\;dx=2\int _{-c}^{0}\cos {x}\;dx=2\sin {c}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43f875b81a6b5c38d2d0a7f54fbe9ae958b47526)
![{\displaystyle \int _{-c}^{c}\tan {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a562f35f07612ecdd0e562ba5e93b728320032)
Integral en un cercle complet
![{\displaystyle \int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{2n+1}{x}\,dx=0\!\qquad n,m\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/da2cb438081529ad3bb1eb8c502d8ae9cb6cc564)
Referències
- ↑ Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008
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