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Die Konstante
wird als ungleich 0 angenommen, und die Integrationskonstante wurde weggelassen.
Integrale trigonometrischer Funktionen, die sin enthalten
[Bearbeiten]
![{\displaystyle \int \sin cx\;dx=-{\frac {1}{c}}\cos cx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90927a9524ca2ebb7532b3b642442416f997074c)
![{\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{( }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/466d0c15ca0f9314b8e4ea155410e05f3a97660b)
![{\displaystyle \int {\sqrt {1-\sin {x}}}\,dx=2{\frac {\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}{\sqrt {1-\sin {x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41c875867982adf8ab314417d6faeb96a6d34a2a)
![{\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}}}-{\frac {x\cos cx}{c}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba196bb420de05f3a536e722b9fa9b633ae1edb)
![{\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{( }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85fd8899dc119d4fd51b3d062b5b611cae4b2765)
![{\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{( }}n=2,4,6...{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a73c1c88b847bec210d6af7825627943ca00787f)
![{\displaystyle \int {\frac {\sin 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/d4c679a50c11f75c5624254025badb770884691f)
![{\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\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2c4bcf48fc3ff6ee0d8eb14b330bf55dacd67f)
![{\displaystyle \int {\frac {dx}{\sin cx}}={\frac {1}{c}}\ln \left|\tan {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33e9b1d708e4a982d400c2bf4ce60212c69b1273)
![{\displaystyle \int {\frac {dx}{\sin ^{n}cx}}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad {\mbox{( }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b9283e5fb23994b2a182abad3af4409f0954ce)
![{\displaystyle \int {\frac {dx}{1\pm \sin cx}}={\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce341569f2cd1bf50b4d093c9a4a5c6f236505ac)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ad92be2fe75f28b3a47ea3948950df8e6efcb4)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b478db3bb323ac13e34688e726f9c5b72fb44d)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e1d0c0992f04eb68ac1d4b84f8c8fd96e43064)
![{\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin \left((c_{1}-c_{2})x\right)}{2(c_{1}-c_{2})}}-{\frac {\sin \left((c_{1}+c_{2})x\right)}{2(c_{1}+c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7971aeaa258603fd2aeff0bf2e8a444493d174c)
Integrale trigonometrischer Funktionen, die cos enthalten
[Bearbeiten]
![{\displaystyle \int \cos cx\;dx={\frac {1}{c}}\sin cx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5a071bc63ba7f827d559d720b4e5ef34b64d48)
![{\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{( }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cac03b56479930465b5d966be22de3c1cab5659)
![{\displaystyle \int x\cos cx\;dx={\frac {\cos cx}{c^{2}}}+{\frac {x\sin cx}{c}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5fc103923703bd335b87b1bffc18e6ed72832e)
![{\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/593a0631f1f6c5d29a40eed7e6035a515ff8e3d6)
![{\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{( }}n=1,3,5...{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f9aa6f706b9725eb4817f4e6a7a87bd7f78ee1)
![{\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/babee406174e7983169482bebe8d2f7b8f05a2f6)
![{\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{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acd1a2721b8a02e6127b4c0152da6a00a17e1529)
![{\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 {\sin cx}{c(n-1)cos^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{( }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7d6581ac7f29fd7d8d5941cd5d97a2c6dc0f1c)
![{\displaystyle \int {\frac {dx}{1-\cos cx}}=-{\frac {1}{c}}\cot {\frac {cx}{2}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24e320bd9d4ec7e20248bbb72a2ab04ea060355c)
![{\displaystyle \int {\frac {x\;dx}{1+\cos cx}}={\frac {x}{c}}\tan {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\cos {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d18ff5f4712370e7b333e04b0220acba13840b0)
![{\displaystyle \int {\frac {x\;dx}{1-\cos cx}}=-{\frac {x}{c}}\cot {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\sin {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62d94b8fcbba8f323b79bc40238f50ced144f145)
![{\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/b785bd70c86d08bbb26514f8f2e5d5302886c140)
![{\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/c5924bce1ec5d6eae2f8ff87b166bea8199d3b46)
![{\displaystyle \int \cos c_{1}x\cos c_{2}x\;dx={\frac {\sin \left((c_{1}-c_{2})x\right)}{2(c_{1}-c_{2})}}+{\frac {\sin \left((c_{1}+c_{2})x\right)}{2(c_{1}+c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f33ad17d9907df0c742d14052201baa425c9408)
Integrale trigonometrischer Funktionen, die tan enthalten
[Bearbeiten]
![{\displaystyle \int \tan cx\;dx=-{\frac {1}{c}}\ln |\cos cx|\,\!={\frac {1}{c}}\ln |\sec cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c913537b6e079a80ab0caa4b24d4cf18b14cd9d)
![{\displaystyle \int \tan ^{n}cx\;dx={\frac {1}{c(n-1)}}\tan ^{n-1}cx-\int \tan ^{n-2}cx\;dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c057f8032ddb285c3c534809e62ad1d3c3fa5b3d)
![{\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/378ca3a1ec5d434aabbdf51e48373878fd21bd55)
![{\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/11ba73cb9cdef52d2fc4527075cea0310344c7e9)
![{\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/6e44adbe3d6c8dcb668a2510e034eb472255ab69)
![{\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/69ef7342905057f0b5d5af57d0a472c818529d51)
Integrale trigonometrischer Funktionen, die sec enthalten
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![{\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{ ( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d82c4e6652b89c81226f7395c0b9e538c9098314)
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcc12662b9bdd3e263861867ebee100e8d244d7)
Integrale trigonometrischer Funktionen, die csc enthalten
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![{\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/8e2aa8997fe046d1daad90c3b401c1905a539469)
![{\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{ ( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3bf9348ae0aa7facb3c7cad62890069c5b74a8e)
Integrale trigonometrischer Funktionen, die cot enthalten
[Bearbeiten]
![{\displaystyle \int \cot cx\;dx={\frac {1}{c}}\ln |\sin cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3d36717ab57ddfb45a6d98f9ee5a7264aaa8d1)
![{\displaystyle \int \cot ^{n}cx\;dx=-{\frac {1}{c(n-1)}}\cot ^{n-1}cx-\int \cot ^{n-2}cx\;dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b98b405e7b645a8080df2a391790c47732353c1d)
![{\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/011756769be171392a20919b191f9ff30349d68b)
![{\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/0e9f3f2697d4e65fdd9c46ae47c080f0fbee0cee)
Integrale trigonometrischer Funktionen, die sowohl sin als auch cos enthalten
[Bearbeiten]
![{\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+\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 cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7c28a3cbb8faae8da164304c18ffac19bd99cd)
![{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/641492bf3f2102ff0e0a3c3aab3fadb29fd74c4e)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948e3466f7b59d4a7e6e34fa5c4a5f6536ad7420)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/635f3cb11427290dd7b1146eea072bd34274c244)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb14e2970b623def16c9fef2abcaed2230b4981)
![{\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|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cb83257ae9be351676972155dcd06c9f77ad00)
![{\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/5b23856c5dc1d08f8df89537a8199d6cc6e631b2)
![{\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{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fcf57b17e9d46e74d55a94319b42e07e3b3d1d7)
![{\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82ed43753482d7f4f9203c866999bd03417494a0)
![{\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8364b79ca100f1c2495b2d0498f3d337ee6bc83)
![{\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{( }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88a26cd522e2f07da8ea2e479dcaa5223297c876)
- auch:
![{\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{( }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a28de0422c56b03f9dede1f1818630db1bb2ef92)
![{\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{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa55e16bb7ae2475fe29834529727df4413014f3)
![{\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{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/737434945c66cd0d4b146d07de294a71dd3364b1)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/218083942f7cd58e7dc1997647c62595fb1ec9e2)
![{\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{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c6dad4787a8dc281f8451ebe694df2ccf40418)
![{\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{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8695d54f1af4201bbe882ad2303c3145ead92056)
![{\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{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98b1c5f14597750be6b90ee7b919c57485ed56ca)
- auch:
![{\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{( }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed219a4f688c0586cdb990abfd3df77a8540a38b)
- auch:
![{\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{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cba8ecca29dcbe15eb35634fbb597d935af392e1)
![{\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a32a5742c1035350af73ac78e10e7f25760f2e8)
![{\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{( }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/416ee00d79c1f9bf5a30b89bd794c99014c2fe4a)
![{\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{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eca6226eeb75a120832c81cd8d4b3ef08678f0d)
- auch:
![{\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{( }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/264a025ca07841c7ec58f6d9015726bf5eda2b12)
- auch:
![{\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{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5687ad6dd6bcce37b9605c98a53580687e1f45)
Integrale trigonometrischer Funktionen, die sowohl sin als auch tan enthalten
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![{\displaystyle \int \sin cx\tan cx\;dx={\frac {1}{c}}(\ln |\sec cx+\tan cx|-\sin cx)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70a803de5cb5fb6444c37f130c7faf1b10db21cc)
![{\displaystyle \int {\frac {\tan ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n-1)}}\tan ^{n-1}(cx)\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8f3cd583cb99ec8d94d9eff721d2bda0a68f53)
Integrale trigonometrischer Funktionen, die sowohl cos als auch tan enthalten
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![{\displaystyle \int {\frac {\tan ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\tan ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8e9d3dbc4da925fd8c111e2c88732aa28f35889)
Integrale trigonometrischer Funktionen, die sowohl sin als auch cot enthalten
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![{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n+1)}}\cot ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b650cd74ba3fbb2521acd3f71e331569903f26)
Integrale trigonometrischer Funktionen, die sowohl cos als auch cot enthalten
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![{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\tan ^{1-n}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e443c27bae0142ff033bfb52ea92c14ef3028896)
Integrale trigonometrischer Funktionen, die sowohl tan als auch cot enthalten
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![{\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{( }}m+n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9bb5c65b51c1ad35705c123a3472a4d4ed73bb)
Integrale trigonometrischer Funktionen mit symmetrischen Grenzen
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![{\displaystyle \int _{-c}^{c}\sin {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33469f374c9c3af903d9607671dcfdf18c1a5077)