Inverse Trigonometric Functions
| Formula | Integral |
|---|---|
| \displaystyle {\int_a^b f(x)\dfrac{\mathrm dg(x)}{\mathrm dx} \mathrm dx} | \displaystyle {- \int_a^b \dfrac{\mathrm df(x)}{\mathrm dx} g(x) \mathrm dx + f(x)g(x)\bigg|_a^b} |
| Formula | Integral |
|---|---|
| \displaystyle \int a\,\mathrm dx | ax+C |
| \displaystyle \int x^{n}\,\mathrm dx | {\dfrac {x^{n+1}}{n+1}}+C\quad x \ne 1 |
| \displaystyle \int (ax+b)^{n}\,\mathrm dx | {\dfrac {(ax+b)^{n+1}}{a(n+1)}}+C\quad x \ne 1 |
| \displaystyle \int {1 \over x}\,\mathrm dx | \ln \left|x\right| + C |
| \displaystyle \int e^{ax}\, \mathrm dx | {\dfrac {1}{a}}e^{ax}+C |
| \displaystyle \int f'(x)e^{f(x)}\, \mathrm dx | e^{f(x)}+C |
| \displaystyle \int a^{x}\, \mathrm dx | {\dfrac {a^{x}}{\ln a}}+C |
| \displaystyle \int {e^{x}\left(f\left(x\right)+f'\left(x\right)\right)\, \mathrm dx} | e^{x}f\left(x\right)+C |
| \displaystyle \int {e^{x}\left(f\left(x\right)-\left(-1\right)^{n}{\dfrac {d^{n}f\left(x\right)}{dx^{n}}}\right)\, \mathrm dx} | e^{x}\displaystyle \sum_{k=1}^{n}{\left(-1\right)^{k-1}{\dfrac {d^{k-1}f\left(x\right)}{dx^{k-1}}}}+C \quad n \in \mathbb N |
| \displaystyle \int {e^{-x}\left(f\left(x\right)-{\dfrac {d^{n}f\left(x\right)}{dx^{n}}}\right)\, \mathrm dx} | -e^{-x}\displaystyle \sum_{k=1}^{n}{\dfrac {d^{k-1}f\left(x\right)}{dx^{k-1}}}+C \quad n \in \mathbb N |
| \displaystyle \int \ln x\, \mathrm dx | x\ln x-x+C=x(\ln x-1)+C |
| \displaystyle \int \log _{a}x\, \mathrm dx | x\log _{a}x-{\dfrac {x}{\ln a}}+C={\dfrac {x}{\ln a}}(\ln x-1)+C |
| Formula | Integral |
|---|---|
| \displaystyle \int a\,\mathrm dx | ax+C |
| \displaystyle \int \sin\left(x\right)\, \mathrm dx | -\cos\left(x\right)+C |
| \displaystyle \int \cos\left(x\right)\, \mathrm dx | \sin\left(x\right)+C |
| \displaystyle \int \tan\left(x\right)\, \mathrm dx | \ln {\left|\sec\left(x\right)\right|}+C=-\ln {\left|\cos\left(x\right)\right|}+C |
| \displaystyle \int \cot\left(x\right)\, \mathrm dx | -\ln {\left|\csc \left(x\right)\right|}+C=\ln {\left|\sin\left(x\right)\right|}+C |
| \displaystyle \int \sec\left(x\right)\, \mathrm dx | \ln {\left|\sec\left(x\right)+\tan\left(x\right)\right|}+C=\ln \left|\tan \left({\dfrac {x}{2}}+{\dfrac {\pi }{4}}\right)\right|+C |
| \displaystyle \int \csc \left(x\right)\, \mathrm dx | -\ln {\left|\csc \left(x\right)+\cot\left(x\right)\right|}+C=\ln {\left|\csc \left(x\right)-\cot\left(x\right)\right|}+C=\ln {\left|\tan {\dfrac {x}{2}}\right|}+C |
| \displaystyle \int \sec ^{2}\left(x\right)\, \mathrm dx | \tan \left(x\right)+C |
| \displaystyle \int \csc ^{2}\left(x\right)\, \mathrm dx | -\cot \left(x\right)+C |
| \displaystyle \int \sec\left(x\right)\,\tan\left(x\right)\, \mathrm dx | \sec\left(x\right)+C |
| \displaystyle \int \csc \left(x\right)\,\cot\left(x\right)\, \mathrm dx | -\csc \left(x\right)+C |
| \displaystyle \int \sin ^{2}x\, \mathrm dx | {\dfrac {1}{2}}\left(x-{\dfrac {\sin \left(2x\right)}{2}}\right)+C={\dfrac {1}{2}}(x-\sin \left(x\right)\cos \left(x\right))+C |
| \displaystyle \int \cos ^{2}x\, \mathrm dx | {\dfrac {1}{2}}\left(x+{\dfrac {\sin \left(2x\right)}{2}}\right)+C={\dfrac {1}{2}}(x+\sin \left(x\right)\cos \left(x\right))+C |
| \displaystyle \int \tan ^{2}\left(x\right)\, \mathrm dx | \tan \left(x\right)-x+C |
| \displaystyle \int \cot ^{2}\left(x\right)\, \mathrm dx | -\cot \left(x\right)-x+C |
| \displaystyle \int \sin ^{n}\left(x\right)\, \mathrm dx | -{\dfrac {\sin ^{n-1}\left(x\right)\cos\left(x\right)}{n}}+{\dfrac {n-1}{n}}\displaystyle \int \sin ^{n-2}\left(x\right)\, \mathrm dx |
| \displaystyle \int \cos ^{n}\left(x\right)\, \mathrm dx | {\dfrac {\cos ^{n-1}\left(x\right)\sin\left(x\right)}{n}}+{\dfrac {n-1}{n}}\displaystyle \int \cos ^{n-2}\left(x\right)\, \mathrm dx |
| Formula | Integral |
|---|---|
| \displaystyle \int \arcsin\left(x\right)\, \mathrm dx | x\arcsin\left(x\right)+{\sqrt {1-x^{2}}}+C, \quad |x| \leq 1 |
| \displaystyle \int \arccos\left(x\right)\, \mathrm dx | x\arccos\left(x\right)-{\sqrt {1-x^{2}}}+C, \quad |x| \leq 1 |
| \displaystyle \int \arctan\left(x\right)\, \mathrm dx | x\arctan\left(x\right)-{\dfrac {1}{2}}\ln {\vert 1+x^{2}\vert }+C |
| \displaystyle \int \operatorname {arccot}\left(x\right)\, \mathrm dx | x\operatorname {arccot}\left(x\right)+{\dfrac {1}{2}}\ln {\vert 1+x^{2}\vert }+C |
| \displaystyle \int \operatorname {arcsec}\left(x\right)\, \mathrm dx | x\operatorname {arcsec}\left(x\right)-\ln \left\vert x\,\left(1+{\sqrt {1-x^{-2}}}\,\right)\right\vert +C,\quad |x| \ge 1 |
| \displaystyle \int \operatorname {arccsc}\left(x\right)\, \mathrm dx | x\operatorname {arccsc}\left(x\right)+\ln \left\vert x\,\left(1+{\sqrt {1-x^{-2}}}\,\right)\right\vert +C, \quad |x| \ge 1 |
| Formula | Integral |
|---|---|
| \displaystyle \int \sinh \left(x\right)\, \mathrm dx | \cosh \left(x\right)+C |
| \displaystyle \int \cosh \left(x\right)\, \mathrm dx | \sinh \left(x\right)+C |
| \displaystyle \int \tanh \left(x\right)\, \mathrm dx | \ln \,(\cosh \left(x\right))+C |
| \displaystyle \int \coth \left(x\right)\, \mathrm dx | \ln |\sinh \left(x\right)|+C, \quad x\neq 0 |
| \displaystyle \int \operatorname {sech}\left(x\right)\, \mathrm dx | \arctan \,(\sinh \left(x\right))+C |
| \displaystyle \int \operatorname {csch} \left(x\right) , \mathrm dx | \ln |\operatorname {coth}\left(x\right)-\operatorname {csch} \left(x\right)|+C=\ln \left|\tanh \left(\dfrac{x}{2}\right)\right|+C, \quad x\neq 0 |
| Formula | Integral |
|---|---|
| \displaystyle \int \operatorname {arcsinh} \left(x\right)\, \mathrm dx | x\,\operatorname {arcsinh} \left(x\right)-{\sqrt {x^{2}+1}}+C |
| \displaystyle \int \operatorname {arccosh} \left(x\right)\, \mathrm dx | x\,\operatorname {arccosh} \left(x\right)-{\sqrt {x^{2}-1}}+C, x\geq 1 |
| \displaystyle \int \operatorname {arctanh} \left(x\right)\, \mathrm dx | x\,\operatorname {arctanh} \left(x\right)+{\dfrac {\ln \left(\,1-x^{2}\right)}{2}}+C, |x| <1 |
| \displaystyle \int \operatorname {arccoth} \left(x\right)\, \mathrm dx | x\,\operatorname {arccoth} \left(x\right)+{\dfrac {\ln \left(x^{2}-1\right)}{2}}+C,\quad |x| >1 |
| \displaystyle \int \operatorname {arcsech} \left(x\right)\, \mathrm dx | x\,\operatorname {arcsech} \left(x\right)+\arcsin \left(x\right)+C, \quad 0<x\leq 1 |
| \displaystyle \int \operatorname {arccsch} \left(x\right)\, \mathrm dx | x\,\operatorname {arccsch} \left(x\right)+\vert \operatorname {arcsinh} \left(x\right)\vert +C,\quad x\neq 0 |