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![{\displaystyle \int c\cdot f(x)\mathrm {d} x=c\cdot \int f(x)\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b826c60857d47c3a8a902cd629ad9b025505e4)
![{\displaystyle \int {\big (}f(x)\pm g(x){\big )}\mathrm {d} x=\int f(x)\mathrm {d} x\pm \int g(x)\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c09971c0bbffbdefe1cd19d52ac453530bc18246)
where ![{\displaystyle F'=f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c10131b70a7b0b4c39667f5386b14c449a5217e7)
![{\displaystyle \int u\,dv=uv-\int v\,du}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bddb2871f48408d699da1d94af2076a15008989a)
![{\displaystyle \int \mathrm {d} x=x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/846e0a2c0fe686e7c79012b387450561e19bffa7)
![{\displaystyle \int a\,\mathrm {d} x=ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/706e86fb3a7da72390e6834ce3550c6968257f09)
![{\displaystyle \int x^{n}\mathrm {d} x={\frac {x^{n+1}}{n+1}}+C\qquad ({\text{for }}n\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/018e4cd470f8fc8448d4c8051dd4d728a063d956)
![{\displaystyle \int {\frac {\mathrm {d} x}{x}}=\ln |x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f92b9687669fde59e6df06f8bda2356965b4d01)
![{\displaystyle \int {\frac {\mathrm {d} x}{ax+b}}={\frac {\ln |ax+b|}{a}}+C\qquad ({\text{for }}a\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d0e3285d485d99a21224b795eb3779cfde2b1b)
![{\displaystyle \int \sin(x)\mathrm {d} x=-\cos(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb25b0fd7c30b857b31954d12341ebbc2ecd981)
![{\displaystyle \int \cos(x)\mathrm {d} x=\sin(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69448df25c6298c8ff8cfd9f4c499aa9c279b412)
![{\displaystyle \int \tan(x)\mathrm {d} x=\ln |\sec(x)|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/763893a3e5b93d3309224b03ec59b2ae4233ce86)
![{\displaystyle \int \sin ^{2}(x)\mathrm {d} x=\int {\frac {1-\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}-{\frac {\sin(2x)}{4}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b20b4dbd210f350db2f003caa73546070fe86b82)
![{\displaystyle \int \cos ^{2}(x)\mathrm {d} x=\int {\frac {1+\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}+{\frac {\sin(2x)}{4}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d834976077cf531da1082caeb6ff9e5a643edd0)
![{\displaystyle \int \tan ^{2}(x)\mathrm {d} x=\tan(x)-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29b7154b1c4c2c8e372b0196875492764fd36980)
![{\displaystyle \int \sec(x)\mathrm {d} x=\ln {\Big |}\sec(x)+\tan(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|+C=2\mathrm {artanh} \left(\tan \left({\frac {x}{2}}\right)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/815e30efeaa833bd35a267e590276f9578972f48)
![{\displaystyle \int \csc(x)\mathrm {d} x=-\ln {\Big |}\csc(x)+\cot(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62cee13d9b38f5e5d721c3d8a3be8a0fb59fcd77)
![{\displaystyle \int \cot(x)\mathrm {d} x=\ln |\sin(x)|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6161a60cd4da28e9d73335a5a099004264864f4)
![{\displaystyle \int \sec ^{2}(ax)\mathrm {d} x={\frac {\tan(ax)}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6422039517eacdc89d9b3080e2ed0d8f1178df02)
![{\displaystyle \int \csc ^{2}(ax)\mathrm {d} x=-{\frac {\cot(ax)}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9407edd91fd0ff048cbd97ae77347280588735f)
![{\displaystyle \int \cot ^{2}(ax)\mathrm {d} x=-x-{\frac {\cot(ax)}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d860269b1179f51617a7732d621067b7fdc67de)
![{\displaystyle \int \sec(x)\tan(x)\mathrm {d} x=\sec(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/944d633bad78355f3e89028b2d82086ce86f6e7a)
![{\displaystyle \int \sec(x)\csc(x)\mathrm {d} x=\ln |\tan(x)|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/822270729838928ba5ab103ed2c0a67a02bfe5a4)
![{\displaystyle \int \sin ^{n}(x)\mathrm {d} x=-{\frac {\sin ^{n-1}(x)\cos(x)}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d654eda9b984a5d6a5dba8c9aa17b5a84236e1)
![{\displaystyle \int \cos ^{n}(x)\mathrm {d} x=-{\frac {\cos ^{n-1}(x)\sin(x)}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d10dd51d7443b44df5cf38dfa808271169549c0)
![{\displaystyle \int \tan ^{n}(x)\mathrm {d} x={\frac {\tan ^{n-1}(x)}{(n-1)}}-\int \tan ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a17c1c8906775fc7b62370b3d6ea1071eba59c83)
![{\displaystyle \int \sec ^{n}(x)\mathrm {d} x={\frac {\sec ^{n-1}(x)\sin(x)}{n-1}}+{\frac {n-2}{n-1}}\int \sec ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7951f2e801a04a1b4dada8317fd00b95f4112763)
![{\displaystyle \int \csc ^{n}(x)\mathrm {d} x=-{\frac {\csc ^{n-1}(x)\cos(x)}{n-1}}+{\frac {n-2}{n-1}}\int \csc ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64f4f049d10efba03092bc645026d550c6a13bee)
![{\displaystyle \int \cot ^{n}(x)\mathrm {d} x=-{\frac {\cot ^{n-1}(x)}{n-1}}-\int \cot ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae08f5fb69caa5b1f5c34ffa2cdda187a02e247)
![{\displaystyle a^{2}\int x^{n}\sin(ax)\mathrm {d} x=nx^{n-1}\sin(ax)-ax^{n}\cos(ax)-n(n-1)\int x^{n-2}\sin(ax)\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10041a2b83da86064435aac2d4304e933adce581)
![{\displaystyle a^{2}\int x^{n}\cos(ax)\mathrm {d} x=ax^{n}\sin(ax)+nx^{n-1}\cos(ax)-n(n-1)\int x^{n-2}\cos(ax)\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3edd1aa03378e7f0cb6dab14f41a29aba352a0e9)
![{\displaystyle \int \sin ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {1-n}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6ab81b3e989d2a2c4fcbb4fe6a40d8d5e5d868)
![{\displaystyle \int \cos ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\mathrm {sgn} (\sin(x))\cos ^{n+1}(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {n+3}{2}};\cos ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56381b65349e1dea1cc6eb341514c7a08b780c03)
![{\displaystyle \int \tan ^{n}(x)\mathrm {d} x={\frac {1}{n+1}}\tan ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\tan ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1feaf951495a3b9282778d4ebd3298c19f8cb50b)
![{\displaystyle \int \csc ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4cf370832fe88c1b730439e7bb0b891f1f4dd75)
![{\displaystyle \int \sec ^{n}(x)\mathrm {d} x=\sin(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\sin ^{2}(x)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dff086b2308805d643cfed7a83ca1caab3826644)
![{\displaystyle \int \cot ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\cot ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\cot ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa6dce8a6d4e8354ea8ddcb29b27c5445212625)
Where
is the hypergeometric function and
is the sign function.
![{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}=\arcsin(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a08d6dba7eab560b88b318f99051f0bdd007636c)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {a^{2}-x^{2}}}}=\arcsin \left({\tfrac {x}{a}}\right)+C\qquad ({\text{for }}a\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2bb52109b80d4134339f1a392424354db31452)
![{\displaystyle \int {\frac {\mathrm {d} x}{1+x^{2}}}=\arctan(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84fb5199b9bbb4d99358c49e4ad0547fc04b1f9e)
![{\displaystyle \int {\frac {\mathrm {d} x}{a^{2}+x^{2}}}={\frac {\arctan \left({\tfrac {x}{a}}\right)}{a}}+C\qquad ({\text{for }}a\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce352c5e6c451cc68582202223cf3f5958fa09f)
![{\displaystyle \int e^{x}\mathrm {d} x=e^{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb018f1f54e2f51c5e728d6f8fd7dedeefc81ff)
![{\displaystyle \int e^{ax}\mathrm {d} x={\frac {e^{ax}}{a}}+C\qquad ({\text{for }}a\neq 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72f3c058df9fc97d4173b7d10e12dee2ef4515b5)
![{\displaystyle \int a^{x}\mathrm {d} x={\frac {a^{x}}{\ln(a)}}+C\qquad ({\text{for }}a>0,a\neq 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dcfeea9b4303f6b51c1d12e1e3ef06b35121a2a)
![{\displaystyle \int \ln(x)\mathrm {d} x=x\ln(x)-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/678483af2c60e520885037ce4c6f291b13727246)
![{\displaystyle \int e^{x}\sin(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)-\cos(x))+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6be0ebec45f6c133b39cc22a5da350bbb58fb042)
![{\displaystyle \int e^{x}\cos(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)+\cos(x))+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52a552237ca03728850646c8c91fa44b0c5537b0)
![{\displaystyle \int x^{n}e^{ax}\mathrm {d} x={\frac {1}{a}}x^{n}e^{ax}-{\frac {n}{a}}\int x^{n-1}e^{ax}\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8b94ee7b78655ca2d2d645a5e4bc1d30e7f321)
![{\displaystyle \int \arcsin(x)\mathrm {d} x=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac286fc2832bd610080bbea185b1dbc71824a868)
![{\displaystyle \int \arccos(x)\mathrm {d} x=x\arccos(x)-{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c9d97649215e955fef24fb788276290d1899fa)
![{\displaystyle \int \arctan(x)\mathrm {d} x=x\arctan(x)-{\frac {1}{2}}\ln |1+x^{2}|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61026cef0e2aeea1a7d3382dfaddad0f4a920b85)
![{\displaystyle \int \operatorname {arccsc}(x)\mathrm {d} x=x\operatorname {arccsc}(x)+\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83ace06f96c5214c7d79748e7b17407d904e8706)
![{\displaystyle \int \operatorname {arcsec}(x)\mathrm {d} x=x\operatorname {arcsec}(x)-\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc614b8d3363343ee98b367d60201b526a621da)
![{\displaystyle \int \operatorname {arccot}(x)\mathrm {d} x=x\operatorname {arccot}(x)+{\frac {1}{2}}\ln |1+x^{2}|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d33b89d6c2b86d27f1cfc2b29994466efa3207b4)
![{\displaystyle \int \sinh(x)\mathrm {d} x=-i\int \sin(ix)\mathrm {d} x=\cos(ix)+C=\cosh(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd00235d958f6af056fad538eaba1e150754ed43)
![{\displaystyle \int \cosh(x)\mathrm {d} x=\int \cos(ix)\mathrm {d} x=-i\sin(ix)+C=\sinh(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47142905e3f5697eaad8b7c8e572abfad87d51e1)
![{\displaystyle \int \tanh(x)\mathrm {d} x=-i\int \tan(ix)\mathrm {d} x=\log \left|\cos(ix)\right|+C=\log \left|\cosh(x)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a5dec47def4cf6f3aa2867d42a996e518ab9abf)
![{\displaystyle \int \mathrm {csch} (x)\mathrm {d} x=i\int \csc(ix)\mathrm {d} x=\log \left|-i\tan \left({\frac {ix}{2}}\right)\right|+C=\log \left|\tanh \left({\frac {x}{2}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77569dce2311dd6372c39081997858c580ef315b)
![{\displaystyle \int \mathrm {sech} (x)\mathrm {d} x=\int \sec(ix)\mathrm {d} x=2\mathrm {artanh} \left(-i\tan \left({\frac {x}{2}}i\right)\right)+C=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b67990dee2c068980e1723fdf8f7b3584e5c47bc)
![{\displaystyle \int \mathrm {coth} (x)\mathrm {d} x=i\int \cot(ix)\mathrm {d} x=\log \left|-i\sin(ix)\right|+C=\log \left|\sinh(x)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1f0b1bd1c0d3e7df2dbef18ffaea1ac7e09b81)
![{\displaystyle \int \mathrm {arsinh} (x)\mathrm {d} x=x\mathrm {arsinh} (x)-{\sqrt {x^{2}+1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1538612534639bb4e361620daba371fd88e132)
![{\displaystyle \int \mathrm {arcosh} (x)\mathrm {d} x=x\mathrm {arcosh} (x)-{\sqrt {x^{2}-1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f092a326d42022de0e35df6d3e22567665d2dee)
![{\displaystyle \int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {artanh} (x)+{\frac {1}{2}}\ln(1-x^{2})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac22379c09ccde350e0fe26c6deefc7d380fc2d)
![{\displaystyle \int \mathrm {arcsch} (x)\mathrm {d} x=x\mathrm {arcsch} (x)+|\mathrm {arsinh} (x)|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f397b43e4c662a6fccb3274c7df41c0aa6b364c)
![{\displaystyle \int \mathrm {arsech} (x)\mathrm {d} x=x\mathrm {arsech} (x)+\arcsin(x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff82b5e395041d5d17108197bccb9b857d340abb)
![{\displaystyle \int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {arcoth} (x)+{\frac {1}{2}}\ln(x^{2}-1)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/badcadc1b78ac86b78461e72bb872e157273bacf)
, where
is the sign function.
, where
is the Riemann zeta function.
![{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\mathrm {d} x={\sqrt {\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a1e23ca8db7d77c23b446a82d54fbaf80a64aa)
, where
is the gamma function.
![{\displaystyle \int _{0}^{\infty }t^{s-1}e^{-t}\mathrm {d} t=\Gamma (s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/298f7bfb4662ece0628f6913426d477ee04f561e)
, where
is the modified Bessel function of the first kind.
![{\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\mathrm {d} x={\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f86d732686fd23dfe2d9fb588d04c993e404ccf)