The following is a list of
Integrals (
Antiderivative functions) of
area 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\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\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>