از ویکیپدیا، دانشنامهٔ آزاد
در ادامه فهرستی از انتگرال تابعهای مثلثاتی نوشته شدهاست. برای آگاهی از انتگرال تابعهای نمایی و مثلثاتی فهرست انتگرال تابعهای نمایی را نگاه کنید، همچنین برای داشتن یک فهرست کامل صفحهٔ فهرست انتگرالها را نگاه کنید.
اگر تابع
را شکل کلی تابع مثلثاتی در نظر بگیریم و
را به عنوان مشتق آن، آنگاه:
![{\displaystyle \int a\cos nx\;dx={\frac {a}{n}}\sin nx+c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78945c2df7d6a892107d050fc88454cb8cc0a778)
در تمامی رابطهها فرض میشود که a ناصفر است و C ثابت انتگرالگیری است.
انتگرالهایی که تنها تابع سینوس دارند[ویرایش]
![{\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 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/6135c64146055c376ed388d20bc027bc2c5611a3)
![{\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/d616d15d370e9f33a475370d8e3c90ba917092a1)
![{\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{(for }}|b_{1}|\neq |b_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff6f29b3f4a9d2f3fcceb4abc8e24b0458f79f9)
![{\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{(for }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c535bd71e400101d02f1a8645805ce121b50dc9)
![{\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{(for }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09eee3ae78aaffcf2218077e18bf1d44316cc49)
![{\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=\sum _{k=0}^{2k\leq n}(-1)^{k+1}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\cos ax+\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-1-2k}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\sin ax\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df53ecfceea8fc0884d2824c441055c0ed3214fa)
![{\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{(for }}n=2,4,6...{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d006ad538fdc542282e62804c82784991ce6474)
![{\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)
انتگرالهایی که تنها تابع کسینوس دارند[ویرایش]
![{\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 ^{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 \cos ^{n}ax\;dx={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f91f2ae3f215256de1e1ba13359ef2ed362e85e6)
![{\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 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/9b3a65fbb1c6a1bfbd3285c43301267ec70202a9)
![{\displaystyle \int x^{n}\cos ax\;dx={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\;dx\,=\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-2k-1}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\cos ax+\sum _{k=0}^{2k\leq n}(-1)^{k}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\sin ax\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7457aa1d1067e0c5b8838fd3824f2d949bf064ad)
![{\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{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9636752d6c1166442664f7ea2f89d479edcb2a0d)
![{\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{(for }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96611566428b28e9c6769593d2f5d2daf73a8701)
![{\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{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f54ef1e5544212dca721d017d78cc22a1e6e1c)
انتگرالهایی که تنها تابع تانژانت دارند[ویرایش]
![{\displaystyle \int \tan ax\;dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |secax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02650ce85e6ad512fac45b7cb625a70fbf0dbfe9)
![{\displaystyle \int \tan ^{n}ax\;dx={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\;dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675893e0e7cc8a5abc9820808630b8c85bc4828c)
![{\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{(for }}p^{2}+q^{2}\neq 0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d1db1df5d4a34f0b9d3f85f9d4c127ad6d77a6)
![{\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)
انتگرالهایی که تنها تابع سکانت دارند[ویرایش]
![{\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/62448efb9e0512c1014643b2efa34928c397f1b0)
![{\displaystyle \int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-2}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b40a7152e34b384abc9e4c55e4fac3711bca10fe)
[۱]
![{\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)
انتگرالهایی که تنها تابع کسکانت دارند[ویرایش]
![{\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-2}{ax}\cot {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d31180af9e006a7bd543d89d96012fcf6f3d4f2)
![{\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)
انتگرالهایی که تنها تابع کتانژانت دارند[ویرایش]
![{\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{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/670d900ec7880388c6559ee13b3aefddfdade0d4)
![{\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)
![{\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/49cae478476f6ceb2204cf08f2ac9d4ec88e5018)
![{\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}}\cos ^{2}ax+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e6f00877b30557b9ed59d716ac9bdb00aba084)
![{\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{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e256f8f1b384b14127a62dbc4fd82868e21dd0d7)
![{\displaystyle \int \sin ^{n}ax\cos ax\;dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99084aaa91d621e36f44e2111d68e297292e5cb6)
![{\displaystyle \int \sin ax\cos ^{n}ax\;dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2876501394860452dc0deb008172df04b0a58b37)
![{\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{(for }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26448530bcdee6298cc5d6e7567e0eec874ccf50)
![{\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{(for }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72a4921a8158da092a461586d9707298798ce621)
![{\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{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4a474a9f8de691390d759386e64ea4fc565e2c)
![{\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{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4dd96529744651a07fadb522c829a10e85ebfec)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cca5dbe35552222ebe9d0ca5eb6c45b2c8673c21)
![{\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{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65cd5c8d41f0631bf80850e04fe2aa7743e0ebdb)
![{\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{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce5e65b17891b3826d834fa46343d984f9b9720)
![{\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{(for }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db50b023be04b08e34f6433d144ca77124b712e8)
![{\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{(for }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4afc86b39a46ec53eec57563f38ae9ab6689fa)
![{\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{(for }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d03c99939f7745460be74407c77c482d7849dc9)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03484f4e54b414a9bb395997b9dea7cecb42cf23)
![{\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{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1249ed03d747413cffbf19add52d0cbf22e7435)
![{\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{(for }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67de023409f7600f6cbece947fd869df3cd5bd91)
![{\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{(for }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f27baa52dacd3183439a872537f93374fe10725)
![{\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{(for }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2ba1c43d792d05d847297e1c01a1edfac451b0)
![{\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{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef089d831120034730643e2bdb37d21aec47e3bb)
![{\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f57f211ee99b5111b38508c2250026aad8f41063)
![{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\sin ^{2}ax}}={\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a396fe82730301e77ec40b9ce60a4d9fc224ceb)
![{\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc10ed08ff94acc57c1c4fcd010554af4f69040)
انتگرالهای با بازههای متقارن[ویرایش]
![{\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)
![{\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{(for }}n=1,3,5...{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c27f64b02e1b9319f1438ef3881331cc393a4c4c)
- ↑ Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008