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Dit artikel bevat een lijst van integralen van goniometrische functies. Goniometrische functies zijn in de goniometrie gedefinieerde functies. Integralen zijn het onderwerp van studie van de integraalrekening.
![{\displaystyle \int \sin ax\;dx=-{\frac {1}{a}}\cos ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ead1bbe4346512407abb4ab6f5a786245feb4ce)
![{\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/2bf3f3073986b5500adb3df4ed9b86286594b4ba)
![{\displaystyle \int x\sin ^{2}{ax}\;dx={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f48650cbbd46c69b341104f505ac62d36170b8a4)
![{\displaystyle \int x^{2}\sin ^{2}{ax}\;dx={\frac {x^{3}}{6}}-\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax-{\frac {x}{4a^{2}}}\cos 2ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7beeedaf0dd716b2e3bfe5a8b9c175eb6a1802f9)
![{\displaystyle \int \sin b_{1}x\sin b_{2}x\;dx={\frac {\sin((b_{1}-b_{2})x)}{2(b_{1}-b_{2})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(voor }}|b_{1}|\neq |b_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72f66416651050b8146e703dc57af833c5690da)
![{\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{(voor }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/409b108481b59c366d19f6b6cccd593aa48d194c)
![{\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{(voor }}n>1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e76c64356b866de09c93fbb6e8c6a6a2e4c8941)
![{\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/68595010eb51c2090127c6ef4056229dc649054e)
![{\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{(voor }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46f4d5bc544e4d1401dca3590d96694ab6dd69c0)
![{\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{(voor }}n=2,4,6...{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eccb61ee50dd6e21cdc6ca13a505d04efac2e8f)
![{\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/77de344b2a80e7c4db0a144ef03bf4a0f6676f83)
![{\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/74aa71054408525d841b65e00e7de7815954fe51)
![{\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)
![{\displaystyle \int \cos ax\;dx={\frac {1}{a}}\sin ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c7117856a5b259f860fb84c39cd5a145c88cb2)
![{\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{(voor }}n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3100651dc3623ce552a6ea65a5da756b5069db25)
![{\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/18ff1c6e9514a78010d6dd83d5ec8af5f169e314)
![{\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/a7cb785ad9cbba710e56e0bc121d4b9e1115bd9a)
![{\displaystyle \int x^{2}\cos ^{2}{ax}\;dx={\frac {x^{3}}{6}}+\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax+{\frac {x}{4a^{2}}}\cos 2ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/146878a62734d375bac7bb42d1889e8f34cfd462)
![{\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/0f6c32cffd4e5cb3f235bb41554ea7d23c732a6a)
![{\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/b3393dd8cea2019a0912287d45fbe1b2b020b1c5)
![{\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{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d91d67bb4291aa469ff5e42ee82703aca6e6a48)
![{\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{(voor }}n>1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2420a78a17c944d88278091a2bdba18a51710354)
![{\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/41a971cb7f555d9f48a9f2b820bcc7fe53f2436c)
![{\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/6a8d0cb833e9a78d8ea6ff57d1ce08c44aaa09c7)
![{\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/60f13d0b92fa92359107032c46a9757bf2aa69d0)
![{\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/3e52ed907ccf185d2b5ad2527e38f2bfc91cf786)
![{\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{(voor }}|a_{1}|\neq |a_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c24290da45a8ebbf0a6355ecc76abc670705459)
![{\displaystyle \int {\sqrt {1+\cos x}}\ dx={\sqrt {2}}\ \int \cos {\frac {1}{2}}x\ dx=2{\sqrt {2}}\ \sin {\frac {1}{2}}x\qquad {\mbox{overgaan naar de halve hoek}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f425d15b01b23e63ff9bcea792a343c1534aa81)
![{\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/15fc95f93e46b9f442150e088597ae4d5307439e)
![{\displaystyle \int \tan ^{n}ax\;dx={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\;dx\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c731f6fc1cecbd6f3eab47857db464900c1ae02)
![{\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{(voor }}p^{2}+q^{2}\neq 0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e841f8d1245f2591332fac3495e4c0402b7a21f)
![{\displaystyle \int {\frac {dx}{\tan ax}}={\frac {1}{a}}\ln |\sin ax|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cec87b7e0bf1782eaa468e52078d3f834144e6b1)
![{\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/5177a72f599e3f7a16ed7e208f5688a7bfd1175d)
![{\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/1ad458ce6b433ead7a78170d1dd49c93f195fbe0)
![{\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/7c25af06c5d9fa98071957c51a865ffd872fe1a7)
![{\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/2387ca3c938e779ccceac4839761017881545c43)
![{\displaystyle \int \cot ax\;dx={\frac {1}{a}}\ln |\sin ax|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a929127fc669643aae5546839144bcf5d4bd5dde)
![{\displaystyle \int \cot ^{n}ax\;dx=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;dx\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1789eceedcd81d28728ff252dbcf20ba6d5cc31c)
![{\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/081d8acd969815d9c82e6016beda99d8736daa8c)
![{\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/5835e54365d7de2a0574d91cd4aec0549ff94f5e)
![{\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 ^{2}{x}\,dx=tan{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4615357b8fccb40609d90a685b997cb830fd3486)
![{\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{ (voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789202a777c2e4bd0a3ee8c794aa4b90c28b76a9)
![{\displaystyle \int \sec ^{n}{x}\,dx={\frac {\sec ^{n-2}{x}\tan {x}}{n-1}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{x}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c22e8160f1fca39c014c254d13a5420a2b5ce8e)
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9acabbd90de19b0d361d572dce3398a57c9d653f)
![{\displaystyle \int {\frac {dx}{\sec {x}-1}}=-x-\cot {\frac {x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2845a0bb3c940ca6f9d98303dd5944618ad6a93c)
![{\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/9bf93e0314926a5b59d4cd9cc2c9b9d1134a369c)
![{\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{ (voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ae9f43696dbd80394cd16059f375f1e6dc31a7)
![{\displaystyle \int {\frac {dx}{\csc {x}+1}}=x-{\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da23438b8ab44a1a5d531d59784c7db1bfc488b8)
![{\displaystyle \int {\frac {dx}{\csc {x}-1}}={\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f78276fae595b97b83c96131d3e596005aa2354)
![{\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/24c30dc4abd3017e33de72896af06497c03dce6f)
![{\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{(voor }}|a_{1}|\neq |a_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0153820c0d51620ef1a8d56edfef189d2295dcd)
![{\displaystyle \int \sin ^{n}ax\cos ax\;dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(voor }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9c385fc7c0eb42bd3127bb3951325c83dc03a9)
![{\displaystyle \int \sin ax\cos ^{n}ax\;dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(voor }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65e34c658c9700b6186f8a450b46e579a9a63ff1)
![{\displaystyle \int \sin ^{2}ax\cos ^{2}ax\;dx={\frac {x}{8}}-{\frac {\sin {4ax}}{32a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8422e81f9686d1fde4bf77036fd6856e3b13f883)
![{\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{(voor }}m,n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffff01080d2f6391056438d7ad5030af16a76aa1)
![{\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{(voor }}m,n>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07035c2fb2547eb8726c5d8e38ca8d4b65fce8ae)
![{\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{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abf719733d409bb97ea2a66f55fb366e1513a255)
![{\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{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ab70ba6efab8d27e44a73aa4d111238506fbc7)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6912476ecc034be9f9722a07bf347d0340ac6499)
![{\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{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92b379f2c773c5501b2aa7f6cf75b7dad4018a60)
![{\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{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f75e380581f70ea63a7d3f94fbc92a7badb354d)
![{\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{(voor }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b5c9f20b070029a32951ac1401b20896f5611a)
![{\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{(voor }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60890a010fbc894867d3998e0e888795c702fb4e)
![{\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{(voor }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/312c85dbaf97eae7859129493742237441260c31)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088ab63a8f1e49e0b63b5e48ff779f3c6f560418)
![{\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{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b63ff1cc8b5a08c0717fb6dfe5204ac31b323fb6)
![{\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{(voor }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/501c3fdf9e49b963612ff389daaf7bc98d38bce0)
![{\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{(voor }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/698cc81db0d66ccfdff65d6cc3a2c4b7e2537f98)
![{\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{(voor }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db6fcbc4e9395de46eb1dddb864c7267934f093e)
![{\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/60a858832ea79dd288a97620dae07d10ec9fec50)
![{\displaystyle \int \sin ^{2}ax\tan ^{2}ax\;dx={\frac {\sin {2ax}}{4a}}-{\frac {\tan {ax}}{a}}-{\frac {3x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e550b2a8137b43b20c1bc3fd4c0df75a1ffa82)
![{\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bfcbdfe67c0411027dec31117bc985df0ffe07)
![{\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(voor }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2105158c222fb2bf4dc50488dfae29361346140)
![{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\sin ^{2}ax}}={\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(voor }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88f21ed6ce51d05a87f4e274ae2e35b9ee4beb4b)
![{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a593f0ffedfa16fa508b1d53b67e7cf94d354670)
Goniometrische integralen met symmetrische grenzen[bewerken | brontekst bewerken]
![{\displaystyle \int _{-c}^{c}\sin {x}\;dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99172209bfb37953d66d12a8b011fa92a578462f)
![{\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/ad48dd2f28e38703ac5aae0b5a3d3d01a94bade2)
![{\displaystyle \int _{-c}^{c}\tan {x}\;dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83255e4d8958dd8fd72477d4abb3d2b581b8ae81)
![{\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{(voor }}n=1,3,5...{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8de2f163ed290a0192a351d395336e32e32d91aa)