Encyclopedia > List of integrals of irrational functions

  Article Content

List of integrals of irrational functions

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^2dx}{\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{arsinh}\frac{x}{a}\right)</math>

<math>\int x\sqrt{x^2+a^2}dx=\sqrt{1}{3}\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{arsinh}\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{arsinh}\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{arsinh}\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}\mp a^2\,\mathrm{arcosh}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge\|a|\mbox{; }-\mbox{ for }x>0\mbox{, }+\mbox{ for }x<0\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{arcosh}\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{arcosh}\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{arsinh}\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>



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
Sakhalin

... the period during which vegetation can grow averages less than 100 days. Fishing is actively prosecuted. History Sakhalin, whose natives made contact with the Manchu ...

 
 
 
This page was created in 36.5 ms