Encyclopedia > List of Integrals (irrational functions)

  Article Content

List of integrals of irrational functions

Redirected from List of Integrals (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
Dynabee

... the axis, and the gyroscope's evasive action towards the externally applied force will cause one end of the axis to push against the upper rim of the groove, while the ...

 
 
 
This page was created in 29.3 ms