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
French resistance

... June 10 major Otto Dickmann[?]’s troops wiped out the village of Oradour-sur-Glane in retaliation. German intelligence did not give up, either. Hugo Bleicher arrested ...

 
 
 
This page was created in 31.1 ms