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Q15  Integrate the functions \frac{1}{\sqrt {(x-a)( x-b )}}

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(x-a)(x-b) can be written as  x^2-(a+b)x+ab
                                              \\x^2-(a+b)x+ab+\frac{(a+b)^2}{4}-\frac{(a+b)^2}{4}\\ (x-\frac{(a+b)}{2}^2)^2-\frac{(a-b)^2}{4}

\Rightarrow \int\frac{1}{\sqrt{(x-a)(x-b)}}dx=\int \frac{1}{\sqrt{(x-\frac{(a+b)}{2}^2)^2-\frac{(a-b)^2}{4}}}dx
                                                       let
                                                               x-\frac{(a+b)}{2}=t \Rightarrow dx =dt
                                                        So,
                                                               \\=\int \frac{1}{\sqrt{t^2-(\frac{a-b}{2})^2}}dt\\ =\log \left | t+\sqrt{t^2-(\frac{a-b}{2})^2} \right |+C\\ =\log \left | x-(\frac{a+b}{2})+\sqrt{(x-a)(x-b)} \right |+C                    
 

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manish

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