De la Wikipedia, enciclopedia liberă
Următorul articol este o listă de integrale (primitive) de funcții trigonometrice. Pentru o listă cu mai multe integrale, vezi tabel de integrale și lista integralelor.
Unde c este o constantă:
![{\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{(pentru }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bdb789e7c5352f823bdc7da799c1f1fcb70f545)
![{\displaystyle \int \sin ^{2}{cx}\;dx={\frac {x}{2}}-{\frac {1}{4c}}\sin 2cx\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d959d6e5da7851ecb8e31391789ae048f0b2958)
![{\displaystyle \int {\sqrt {1-\sin {x}}}\,dx=\int {\sqrt {\operatorname {cvs} {x}}}\,dx=2{\frac {\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}{\sqrt {\operatorname {cvs} {x}}}=2{\sqrt {1+\sin {x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/644e701f7bfac6170be097c4eda7308c9c8ec26e)
unde cvs{x} este funcția Coversinus
![{\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{(pentru }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd43e54c9c09b27ec1fe3b848199a767926c0f5a)
![{\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{(pentru }}n=2,4,6\dots {\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/795da971c4e4c86ccfd92908a7e12f7a91f5b2ec)
![{\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{(pentru }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe81c738debafaf3361ede41884c3b2d834d47b3)
![{\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(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(pentru }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4eae4a62532af5ce98bf545b35c0c0f4de9d70)
![{\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{(pentru }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d6008528be2d1643789051adcfca51e215f98b)
![{\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{(pentru }}n=1,3,5\dots {\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a75c0ffea2dcfee81d2305d53da836fa01455413)
![{\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{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6127e523f18d73b1f78332b4911f47a83171d963)
![{\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{(for }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/076195608e3d15a1b18b493aff4627b57ef1b4f0)
![{\displaystyle \int {\frac {dx}{1+\cos cx}}={\frac {1}{c}}\tan {\frac {cx}{2}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/478322b4056d8192a77bc75153738e2508453aa6)
![{\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(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(pentru }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/902d3b3dbddf51669d20ab73c2dd06d080f92460)
![{\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 {\frac {dx}{\tan cx}}={\frac {1}{c}}\ln |\sin cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32416692ac0309457fc465631bf6aec9e9381291)
![{\displaystyle \int \tan ^{n}cx\;dx={\frac {1}{c(n-1)}}\tan ^{n-1}cx-\int \tan ^{n-2}cx\;dx\qquad {\mbox{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4052d2e8b3cc27fe712e0cbf26c61bf75d60969)
![{\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)
![{\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{ (pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a04f74b8ad0a46fbc1442d67ad95ea561e3e28f9)
![{\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/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{ (pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/841dacb70680befb3e58dde0102e9ad5a0adfb8f)
![{\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{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99bab6c20ebcf518908ccf695fb4678a10d8fbf1)
![{\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)
![{\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/f73102e54a5b6ea79f032f025a4998a37cc356ca)
![{\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{(pentru }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d02c4ff717455b7cf7cad4bba161781cdcba8a)
![{\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04dad6181e39c358aa0270583c74dc7c9736bce3)
![{\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b92da5d51456aead9eb87a261a63381fd48170)
![{\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{(pentru }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e33387f6a3ca720f4a12c42c891e8ce81fb87b38)
- also:
![{\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{(pentru }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e39cfdce3b163e2307876b5e57e919b4ffd8fbe3)
![{\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{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c87c06e4a12d5ccc8586fe7c10bd34b8ea444d36)
![{\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{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/650bc8ba01e08a21ba47f6753924c374d0deb0a8)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f80f3c3e2c99108ac8539216ec4c318c727b064e)
![{\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{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9acceefe462d6442d0424d75b4216dfd9a7fbc6c)
![{\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{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/711a1c674756d8a205b5b02ffd0535741047543b)
![{\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{(pentru }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5679aff34f81b22fecb9c5481f5e8d2495753567)
- also:
![{\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{(pentru }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2149ba0aac30fea64fef60f1dcd51ca8f20ed56a)
- also:
![{\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{(pentru }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10c0fea717da3edecd76e9c0e9e3afc60c6d944c)
![{\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b809d6e71b773ace32cde0c4435b669e8999056a)
![{\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{(pentru }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc6d48f110b444b20c349c21109817374d7d137)
![{\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{(pentru }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbb2cb5c8aa9bc27b77f86448de81079c3e6c5d)
- also:
![{\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{(pentru }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf86101c8fcdb01810083a88063789d88a70998)
- also:
![{\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{(pentru }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/686e047bbc5dfab448c6d241717a0b413f59cc50)
![{\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{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab41ca1369932abdf815dc738b0a47b664f1c16)
![{\displaystyle \int {\frac {\tan ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\tan ^{n+1}cx\qquad {\mbox{(pentru }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa71292a90465aa061c48bf68d8ca2579636a38)
![{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n+1)}}\cot ^{n+1}cx\qquad {\mbox{(pentru }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f98b0c7bd32711bbff4eefeb1ee301602be72784)
![{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\tan ^{1-n}cx\qquad {\mbox{(pentru }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f0d4d1da7cb11d703d5fb4bf22e2f8dda9b0c3)
![{\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{(pentru }}m+n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db8fdb0ffea7a589a3f18d8feaca8fc5a95553bc)
Integrale de funcții trigonometrice cu limitele simetrice[modificare | modificare sursă]
![{\displaystyle \int _{-c}^{c}\sin {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33469f374c9c3af903d9607671dcfdf18c1a5077)