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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the functions\[\frac{1}{\sqrt{(x-a)(x-b)}}\]

$\begin{array}{1 1} \log\mid\big(x-\frac{(a+b)}{2}\big)+\sqrt{(x-a)(x-b)}\mid \\ \log\mid\big(x+\frac{(a+b)}{2}\big)+\sqrt{(x-a)(x-b)}\mid \\ \log\mid\big(x-\frac{(a+b)}{2}\big)-\sqrt{(x-a)(x-b)}\mid \\ \log\mid\big(x+\frac{(a+b)}{2}\big)-\sqrt{(x-a)(x-b)}\mid\end{array} $

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Toolbox:
  • $\int\frac{dx}{\sqrt{x^2-a^2}}=log\mid x+\sqrt{x^2-a^2}\mid+c.$
Given:$I=\int\frac{dx}{(x-a)(x-b)}.$
 
$\sqrt{(x-a)(x-b)}=\sqrt{x^2-x(a+b)+ab}.$
 
$\qquad\qquad\quad\quad\;\;=\sqrt{[x-\frac{(a+b)}{2}]^2-\big(\frac{a-b}{2}\big)^2}.$
 
Therefore $I=\int\frac{dx}{\sqrt{[x-\frac{(a+b)}{2}]^2-\big(\frac{a-b}{2}\big)^2}}.$
 
On integrating we get,
 
$\;\;\;=log\mid\big(x-\frac{(a+b)}{2}\big)+\sqrt{(x-a)(x-b)}\mid.$
 

 

answered Feb 4, 2013 by sreemathi.v
 
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