From Wikipedia, the free encyclopedia
The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.[1]
Generally, if the function
is any trigonometric function, and
is its derivative,
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
![{\displaystyle \int \sin ax\,dx=-{\frac {1}{a}}\cos ax+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/17155f417f1407848abf8090096e58430a91d17a)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/94366ec919d67a0f06bb6431029d29b096de77bb)
![{\displaystyle \int \sin ^{3}{ax}\,dx={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/da591fb70e83503367cffa5f8df86754f0181d86)
![{\displaystyle \int x\sin ^{2}{ax}\,dx={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e5aedf47e3b2a17b7f494177d2af59abf5546799)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3902717b8ce182f19cd31aba15f8d9a1cfb72f04)
![{\displaystyle \int x\sin ax\,dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7a98562978db18283202b5ee81589cf1c2dc28d4)
![{\displaystyle \int (\sin b_{1}x)(\sin b_{2}x)\,dx={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(for }}|b_{1}|\neq |b_{2}|{\mbox{)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/30af1446096168913a18a6ee6730651dbe1171b0)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0deecb19cfb405b9e63035571068c18e9a6439e4)
![{\displaystyle \int {\frac {dx}{\sin ax}}=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/39a4cc7433bffdc5cb3e9a1e92fb0990988bce7d)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/41545887c45e5c335c5654fd728b68e2b570bdd3)
![{\displaystyle {\begin{aligned}\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\\&=-\sum _{k=0}^{n}{\frac {x^{n-k}}{a^{1+k}}}{\frac {n!}{(n-k)!}}\cos \left(ax+k{\frac {\pi }{2}}\right)\qquad {\mbox{(for }}n>0{\mbox{)}}\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/5ccfd65881e83676b003b2d02c50af8e82045282)
![{\displaystyle \int {\frac {\sin ax}{x}}\,dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7dba87ad697f99f043327f310fe9b7b966fd7943)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/b1dcfdaf90a90cb3d3e38a3865f92682ab04da8e)
![{\displaystyle \int {\sin {\mathrm {(} }{ax}^{2}\mathrm {+} {bx}\mathrm {+} {c}{\mathrm {)} }{dx}}\mathrm {=} \left\{{\begin{aligned}&{{\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\cos \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){S}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\mathrm {+} {\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\sin \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){C}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {>} }\;{0}}\\&{{\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\cos \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){S}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\mathrm {-} {\sqrt {a}}{\sqrt {\frac {\mathit {\pi }}{2}}}\sin \left({\frac {{b}^{2}\mathrm {-} {4}{ac}}{4a}}\right){C}\left({\frac {{2}{ax}\mathrm {+} {b}}{\sqrt {{2}{a}{\mathit {\pi }}}}}\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {<} }\;{0}}\end{aligned}}\right.\;\;{for}\;{a}\diagup \!\!\!\!{\mathrm {=} }{0}{\mathrm {,} }\;{a}{\mathrm {>} }{0}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/56d4d478b00d52c993be6f70c1c1c13d81cbf112)
![{\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}](https://backend.710302.xyz/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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d08dadaff5165e64f02d1efcc3a48ab10c8c8f9e)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2037a630f830c597c4126f076edd449c8967fbea)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1baffb49b75cb47bcc18564e62c50ad40cc37c11)
![{\displaystyle \int \cos ax\,dx={\frac {1}{a}}\sin ax+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/76fe3b3af800a174faece0db14fcdded789dc979)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6d5a154836333fe3188bc000c0cfe80b86fc8915)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/639c6c30dfaf86cd9b2909bb68fc90bf408e1f8d)
![{\displaystyle \int x\cos ax\,dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7d58daa2d9b221f46b811e2a25309b0fcb64c678)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e8a82a134604ff47f5aecb2a44a092592d160dfc)
![{\displaystyle {\begin{aligned}\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\\&=\sum _{k=0}^{n}(-1)^{\lfloor k/2\rfloor }{\frac {x^{n-k}}{a^{1+k}}}{\frac {n!}{(n-k)!}}\cos \left(ax-{\frac {(-1)^{k}+1}{2}}{\frac {\pi }{2}}\right)\\&=\sum _{k=0}^{n}{\frac {x^{n-k}}{a^{1+k}}}{\frac {n!}{(n-k)!}}\sin \left(ax+k{\frac {\pi }{2}}\right)\qquad {\mbox{(for }}n>0{\mbox{)}}\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2952871c166663dab259233754344b795ba1e9b3)
![{\displaystyle \int {\frac {\cos ax}{x}}\,dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6d82b8c7f2081edd47994c4c5600916e3800fb48)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/746c4c93320501bdbf8fe4c10f4ecf86830fd1af)
![{\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://backend.710302.xyz/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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/46d30472f0c6af81bbfb58ba6fe4370a9f7f3c8f)
![{\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C}](https://backend.710302.xyz/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://backend.710302.xyz/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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7d39822052ca12c117f7121ef13f59d7fadd8ace)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c0c71e740a637ad718742a884ab0284c19dcf861)
![{\displaystyle \int {\frac {\cos ax\,dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/868e1340952b82a678a6ca4c964455ffdb51ec09)
![{\displaystyle \int {\frac {\cos ax\,dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/376acd5ba6dda72e5511ca987900ff39f82c3462)
![{\displaystyle \int (\cos a_{1}x)(\cos a_{2}x)\,dx={\frac {\sin((a_{2}-a_{1})x)}{2(a_{2}-a_{1})}}+{\frac {\sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/813b2d66a6b5cbd286ad30d71d441aa57081c0e8)
![{\displaystyle \int \tan ax\,dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/b34ce80d93081408154a153d81d896074b17aae3)
![{\displaystyle \int \tan ^{2}{x}\,dx=\tan {x}-x+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/83b8d69c24a94eed938f2e751572e874aff74f7f)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/925e1c6fdb836817a46dedd56d9ad1ba3dfbd3aa)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7c8618438a2fdaff26687b201395b05458f808c0)
![{\displaystyle \int {\frac {dx}{\tan ax\pm 1}}=\pm {\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax\pm \cos ax|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/b9a9103841527de43578c2e7776aeb81f0fb114a)
![{\displaystyle \int {\frac {\tan ax\,dx}{\tan ax\pm 1}}={\frac {x}{2}}\mp {\frac {1}{2a}}\ln |\sin ax\pm \cos ax|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/eea65135188c69d920b9f24937a482993fed82a9)
![{\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C={\frac {1}{a}}\ln {\left|\tan {\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)}\right|}+C={\frac {1}{a}}\operatorname {artanh} {\left(\sin {ax}\right)}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/747d013893285558422153b2fd7e1aa9784071eb)
![{\displaystyle \int \sec ^{2}{x}\,dx=\tan {x}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/62448efb9e0512c1014643b2efa34928c397f1b0)
![{\displaystyle \int \sec ^{3}{x}\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/5caad7043f7aa20013456e428c64b7fba0df359f)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/bfb83a90e69050a71b631b159f1b641738f85054)
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}](https://backend.710302.xyz/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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2845a0bb3c940ca6f9d98303dd5944618ad6a93c)
![{\displaystyle \int {\frac {\sin {x}}{\cos {x}}}=\int \tan {x}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e390b3943ca5bc6cf15a39dead513aafa7973e3a)
![{\displaystyle \int \csc {ax}\,dx=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C={\frac {1}{a}}\ln {\left|\csc {ax}-\cot {ax}\right|}+C={\frac {1}{a}}\ln {\left|\tan {\left({\frac {ax}{2}}\right)}\right|}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/cc95cac097eea7e31fbb0a49f428dd903d68a25e)
![{\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/417803af6cef8535c9b9ee74f75a20ab4180fac0)
![{\displaystyle \int \csc ^{3}{x}\,dx=-{\frac {1}{2}}\csc x\cot x-{\frac {1}{2}}\ln |\csc x+\cot x|+C=-{\frac {1}{2}}\csc x\cot x+{\frac {1}{2}}\ln |\csc x-\cot x|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c793315af1ab2724bbd7277cf24812703d8023a9)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/fb9e86eda7e5b586050afa0bb690b5e1794af6d7)
![{\displaystyle \int {\frac {dx}{\csc {x}+1}}=x-{\frac {2}{\cot {\frac {x}{2}}+1}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4d3116bd8d583f72077e90f06cf8e867997fdd14)
![{\displaystyle \int {\frac {dx}{\csc {x}-1}}=-x+{\frac {2}{\cot {\frac {x}{2}}-1}}+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/f12f1dc04dd4f01c7df46f36cba6653112da4418)
![{\displaystyle \int \cot ax\,dx={\frac {1}{a}}\ln |\sin ax|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8fd8d33638f05fb0f16334bb90a8aa016dc05bca)
![{\displaystyle \int \cot ^{2}{x}\,dx=-\cot {x}-x+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/36979324717e61101b7119e6b3e995d5ec509d69)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c0aef6b771a3ad83d5d62e6df67887774d6de8ed)
![{\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\,dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/f957b8e698c1a8b55ba500a962f1b183b557889e)
![{\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\,dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6e697e83f4089e2905dd245cb432fd219dcc493d)
An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules.
![{\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://backend.710302.xyz/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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/25e0e3cebd7eac046797eefb5e8be824a6ec6008)
![{\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}}}={\frac {1}{2(n-1)}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}+(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2}}}\right)}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/fac73d6be8f02747aaba1086a025357e0a8a6d15)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/064e4fca8dab302c4a14d713ffec2d193c49e5aa)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/31c1c965e0a049e416b48908b9083d822fcd820d)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2f7081fc3083a1df9044e044c70fb6749e39772f)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/885121414a63e1158cce975732a0140443059683)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4be1a480baee112532376953a0e514953aac55e5)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/5551ba6b1a17809cd93bad200f96d7bd77c41add)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/cba9a5f34c6745abfcdfea7906197167c4bf7fc4)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/89bd9ffed0a439702f128c452dd2e9d363175ef4)
![{\displaystyle \int (\sin ax)(\cos ax)\,dx={\frac {1}{2a}}\sin ^{2}ax+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0bd704958c5a62ac35486a94c701c4d4d4d89ae1)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/92f3a79d9fc7cc05ff77c79d0f9dc0a0c3506c91)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3d738258cd0ad64e8916b1afcdd40bca57eb6a86)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/95a29ec1949b61595a138149da7af1fe9b056c1e)
![{\displaystyle {\begin{aligned}\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{)}}\\&={\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{)}}\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8cc587066f531ba750784dbd9f5b03aeccc67d7f)
![{\displaystyle \int {\frac {dx}{(\sin ax)(\cos ax)}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e03f904edd8835bf1f3b47bce34e30cf3e2fbf32)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6519ac56d6d1811592f964786eb42751d19b6f8f)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/de84da3e83860aa7dd53d3af5679289da711a241)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9b8dded2ee242701ff20afdc86d891f9070f90fd)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0901ab7048597a2d942a8d9ec9251ab116891ac8)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/9a181636833d2d500b4be5c97a0fc58ab37f96fa)
![{\displaystyle {\begin{aligned}\int {\frac {\sin ^{2}x}{1+\cos ^{2}x}}\,dx&={\sqrt {2}}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}}}\right)-x\qquad {\mbox{(for x in}}]-{\frac {\pi }{2}};+{\frac {\pi }{2}}[{\mbox{)}}\\&={\sqrt {2}}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}}}\right)-\operatorname {arctangant} \left(\tan x\right)\qquad {\mbox{(this time x being any real number }}{\mbox{)}}\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/99bc35b310db277a8b20f736913c8178097758b6)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/872384bfd083802c7e5f81edcc55460dde40addf)
![{\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ^{m}ax}}={\begin{cases}{\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}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\{\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}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\-{\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}}&{\mbox{(for }}m\neq n{\mbox{)}}\end{cases}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/58bdd410008a1b627ede0f2b7104ea01b90192f8)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/32b3ed71c00b7fc752cab0ee21b6b106ccaeed96)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/af8d5900b25ed3cda6de1b98479c1fc0d1c30cd9)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/45036dea1d3b23c1be01e446c207f8a2ecfa1bbb)
![{\displaystyle \int {\frac {\cos ^{n}ax\,dx}{\sin ^{m}ax}}={\begin{cases}-{\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}}&{\mbox{(for }}n\neq 1{\mbox{)}}\\-{\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}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\{\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}}&{\mbox{(for }}m\neq n{\mbox{)}}\end{cases}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6d7373dae8faf0f3bcb7f7ddbd6465b952c9d8dc)
![{\displaystyle \int (\sin ax)(\tan ax)\,dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d503c75c8dcbb712f88809162a2e3e20f1ff7b88)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/152292322de9851a1100065e0efe54701f01578b)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e73c5092aac709c9971bb933c438ebc917344a22)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/309b37abeb46abc52b16f4643f6f367beaba5cae)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8addfded4cf47e0543b3217b3415e08576a74e2d)
![{\displaystyle \int (\sec x)(\tan x)\,dx=\sec x+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d3426300c895f6ff40c28455d36d29417d683dee)
![{\displaystyle \int (\csc x)(\cot x)\,dx=-\csc x+C}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/abd49ac7e4242cab5000f8180c53adcd584240f4)
Integrals in a quarter period
[edit]
Using the beta function
one can write
![{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {1}{2}}B\left({\frac {n+1}{2}},{\frac {1}{2}}\right)={\begin{cases}{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {3}{4}}\cdot {\frac {1}{2}}\cdot {\frac {\pi }{2}},&{\text{if }}n{\text{ is even}}\\{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {4}{5}}\cdot {\frac {2}{3}},&{\text{if }}n{\text{ is odd and more than 1}}\\1,&{\text{if }}n=1\end{cases}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/cbd25f478650fdc0b035db3c07583bf45d1098be)
Integrals with symmetric limits
[edit]
![{\displaystyle \int _{-c}^{c}\sin {x}\,dx=0}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6976aecf1b8b7d692492e777f59de99b7b9b8ac1)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d17e7b3ec2a19c12316f01d5b4f639bdaee1cce7)
![{\displaystyle \int _{-c}^{c}\tan {x}\,dx=0}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c1e0a8bab0e106ca691271ae26382c544c64c073)
![{\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://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/23ee4414783a0d33f9ae707bbfdccd206e6ec935)
![{\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(-1)^{n})}{24n^{2}\pi ^{2}}}={\frac {a^{3}}{24}}(1-6{\frac {(-1)^{n}}{n^{2}\pi ^{2}}})\qquad {\mbox{(for }}n=1,2,3,...{\mbox{)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3306ec56ff1a4ef7ab31b0626adefba80fcdc83e)
Integral over a full circle
[edit]
![{\displaystyle \int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{n}{x}\,dx=0\!\qquad n,m\in \mathbb {Z} }](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1448bcac538b7d3f79cd35ba7a9d4dd4e209f281)
![{\displaystyle \int _{0}^{2\pi }\sin ^{m}{x}\cos ^{2n+1}{x}\,dx=0\!\qquad n,m\in \mathbb {Z} }](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1fd076477398fbae83558bce65d5a78edef13200)