List of integrals of arc functions
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The following is a list of integrals (antiderivative functions) of arc functions. For a complete list of Integral functions, please see table of integrals and list of integrals.
- <math>\int\arcsin\frac{x}{c}\,dx = x\arcsin\frac{x}{c} + \sqrt{c^2-x^2}<math>
- <math>\int x \arcsin\frac{x}{c}\,dx = \left(\frac{x^2}{2}-\frac{c^2}{4}\right)\arcsin\frac{x}{c} + \frac{x}{4}\sqrt{c^2-x^2}<math>
- <math>\int x^2 \arcsin\frac{x}{c}\,dx = \frac{x^3}{3}\arcsin\frac{x}{c} + \frac{x^2+2c^2}{9}\sqrt{c^2-x^2}<math>
- <math>\int x^n \sin^{-1}x\,dx = \frac{1}{n+1}\left(x^{n+1}\sin^{-1}x \right.<math>
- <math>\left. + \frac{x^n\sqrt{1 - x^2} - nx^{n-1}\sin^{-1}x}{n-1} + n\int x^{n-2}\sin^{-1}x\,dx\right)<math>
- <math>\int\arccos\frac{x}{c}\,dx = x\arccos\frac{x}{c} - \sqrt{c^2-x^2}<math>
- <math>\int x \arccos\frac{x}{c}\,dx = \left(\frac{x^2}{2}-\frac{c^2}{4}\right)\arccos\frac{x}{c} - \frac{x}{4}\sqrt{c^2-x^2}<math>
- <math>\int x^2 \arccos\frac{x}{c}\,dx = \frac{x^3}{3}\arccos\frac{x}{c} - \frac{x^2+2c^2}{9}\sqrt{c^2-x^2}<math>
- <math>\int\arctan\frac{x}{c}\,dx = x\arctan\frac{x}{c} - \frac{c}{2}\ln(c^2+x^2)<math>
- <math>\int x \arctan\frac{x}{c}\,dx = \frac{c^2+x^2}{2}\arctan\frac{x}{c} - \frac{cx}{2}<math>
- <math>\int x^2 \arctan\frac{x}{c}\,dx = \frac{x^3}{3}\arctan\frac{x}{c} - \frac{cx^2}{6} + \frac{c^3}{6}\ln{c^2+x^2}<math>
- <math>\int x^n \arctan\frac{x}{c}\,dx = \frac{x^{n+1}}{n+1}\arctan\frac{x}{c} - \frac{c}{n+1}\int\frac{x^{n+1} dx}{c^2+x^2} \qquad\mbox{(for }n\neq 1\mbox{)}<math>
- <math>\int \arcsec{\frac{x}{c}}\,dx = x \arcsec{\frac{x}{c}} + \frac{x}{c|x|}\ln{|x \pm \sqrt{x^2 - 1}|}<math>
- <math>\int x\arcsec{x}\,dx\,=\,\frac{1}{2}\left(x^2\arcsec{x} - \sqrt{x^2 - 1}\right) <math>
- <math>\int x^n\arcsec{x}\,dx\,=\,\frac{1}{n+1}\left(x^{n+1}\arcsec{x} - \frac{1}{n}\left(x^{n-1}\sqrt{x^2 - 1}\; \right. \right.<math>
- <math>\left. \left. + (1-n)\left(x^{n-1}\arcsec{x} + (1-n)\int x^{n-2}\arcsec{x}\,dx \right)\right)\right)<math>
- <math>\int\mathrm{arccot}\,\frac{x}{c}\,dx = x\,\mathrm{arccot}\,\frac{x}{c} + \frac{c}{2}\ln(c^2+x^2)<math>
- <math>\int x\,\mathrm{arccot}\,\frac{x}{c}\,dx = \frac{c^2+x^2}{2}\,\mathrm{arccot}\,\frac{x}{c} + \frac{cx}{2}<math>
- <math>\int x^2\,\mathrm{arccot}\,\frac{x}{c}\,dx = \frac{x^3}{3}\,\mathrm{arccot}\,\frac{x}{c} + \frac{cx^2}{6} - \frac{c^3}{6}\ln(c^2+x^2)<math>
- <math>\int x^n\,\mathrm{arccot}\,\frac{x}{c}\,dx = \frac{x^{n+1}}{n+1}\,\mathrm{arccot}\,\frac{x}{c} + \frac{c}{n+1}\int\frac{x^{n+1} dx}{c^2+x^2} \qquad\mbox{(for }n\neq 1\mbox{)}<math>fr:Primitives de fonctions circulaires réciproques
it:Tavola degli integrali indefiniti di funzioni d'arco pl:Całki funkcji arcus