List of integrals of irrational functions - Definition and Overview

The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, please see table of integrals and list of integrals.

<math>\int\sqrt{a^2-x^2}\;dx = \frac{1}{2}\left(x\sqrt{a^2-x^2}+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}<math>

<math>\int x\sqrt{a^2-x^2}\;dx = -\frac{1}{3}\sqrt{(a^2-x^2)^3} \qquad\mbox{(}|x|\leq|a|\mbox{)}<math>

<math>\int\frac{\sqrt{a^2-x^2}\;dx}{x} = \sqrt{a^2-x^2}-a\ln\left|\frac{a+\sqrt{a^2+x^2}}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}<math>

<math>\int\frac{dx}{\sqrt{a^2-x^2}} = \arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}<math>

<math>\int\frac{x^2\;dx}{\sqrt{a^2-x^2}} = -\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}<math>

<math>\int\sqrt{x^2+a^2}\;dx = \frac{1}{2}\left(x\sqrt{x^2+a^2}+a^2\,\mathrm{arcsinh}\frac{x}{a}\right)<math>

<math>\int x\sqrt{x^2+a^2}\;dx=\frac13\sqrt{(x^2+a^2)^3}<math>

<math>\int\frac{\sqrt{x^2+a^2}\;dx}{x} = \sqrt{x^2+a^2}-a\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|<math>

<math>\int\frac{dx}{\sqrt{x^2+a^2}} = \mathrm{arcsinh}\frac{x}{a} = \ln\left|x+\sqrt{x^2+a^2}\right|<math>

<math>\int\frac{x\,dx}{\sqrt{x^2+a^2}} = \sqrt{x^2+a^2}<math>

<math>\int\frac{x^2\;dx}{\sqrt{x^2+a^2}} = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\,\mathrm{arcsinh}\frac{x}{a} = \frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|<math>

<math>\int\frac{dx}{x\sqrt{x^2+a^2}} = -\frac{1}{a}\,\mathrm{arcsinh}\frac{a}{x} = -\frac{1}{a}\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right|<math>

<math>\int\sqrt{x^2-a^2}\;dx = \frac{1}{2}\left(x\sqrt{x^2-a^2}-\sgn x\,\mathrm{arccosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge|a|\mbox{)}<math>

<math>\int x\sqrt{x^2-a^2}\;dx = \frac{1}{3}\sqrt{(x^2-a^2)^3} \qquad\mbox{(for }|x|\ge|a|\mbox{)}<math>

<math>\int\frac{\sqrt{x^2-a^2}\;dx}{x} = \sqrt{x^2-a^2} - a\arccos\frac{a}{x} \qquad\mbox{(for }|x|\ge|a|\mbox{)}<math>

<math>\int\frac{dx}{\sqrt{x^2-a^2}} = \mathrm{arccosh}\frac{x}{a} = \ln\left(|x|+\sqrt{x^2-a^2}\right) \qquad\mbox{(for }|x|>|a|\mbox{)}<math>

<math>\int\frac{x\;dx}{\sqrt{x^2-a^2}} = \sqrt{x^2-a^2} \qquad\mbox{(for }|x|>|a|\mbox{)}<math>

<math>\int\frac{x^2\,dx}{\sqrt{x^2-a^2}} = \frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\,\mathrm{arccosh}\left|\frac{x}{a}\right| = \frac{1}{2}\left(x\sqrt{x^2-a^2}+a^2\ln\left(|x|+\sqrt{x^2-a^2}\right)\right) \qquad\mbox{(for }|x|>|a|\mbox{)}<math>

<math>\int\frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\ln\left|2\sqrt{a(ax^2+bx+c)}+2ax+b\right| \qquad\mbox{(for }a>0\mbox{)}<math>

<math>\int\frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\,\mathrm{arcsinh}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(for }a>0\mbox{, }4ac-b^2>0\mbox{)}<math>

<math>\int\frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\ln|2ax+b| \qquad\mbox{(for }a>0\mbox{, }4ac-b^2=0\mbox{)}<math>

<math>\int\frac{dx}{\sqrt{ax^2+bx+c}} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad\mbox{(for }a<0\mbox{, }4ac-b^2<0\mbox{)}<math>

<math>\int\frac{x\;dx}{\sqrt{ax^2+bx+c}} = \frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int\frac{dx}{\sqrt{ax^2+bx+c}}<math>