Encyclopedia > List of integrals of arc functions

  Article Content

List of integrals of arc functions

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>



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Christiania

... disambiguation page; that is, one that just points to other pages that might otherwise have the same name. If you followed a link here, you might want to go back and fix ...

 
 
 
This page was created in 25.8 ms