Question

Find : $\int \frac{1}{\sqrt{(x-a)(x-b)}}dx$

Answer 1

$=\int \frac{dx}{(x-a)(x-b)}$

$=\int \frac{dx}{\sqrt{x^2-(a+b)x+ab}}$

$=\int \frac{dx}{\sqrt{(x-{\left(\frac{a+b}{2}\right))}^2+\frac{2ab-a^2-b^2-2ab}{4}}}$

Put $x-\left(\frac{a+b}{2}\right)=t$ $dx=dt$

$=\int \frac{dt}{t^2+\left(\frac{-a^2-b^2}{4}\right)}=\int \frac{dt}{t^2-\left(\frac{a^2+b^2}{4}\right)}$

$=log|t+\sqrt{t^2-\left(\frac{a^2+b^2}{4}\right)}|+c$

$=log|x-(\frac{a+b}{2})+\sqrt{(x-(\frac{a+b}{2}){)}^2-(\frac{a^2+b^2}{4})}|+c$

$=log|x-(\frac{a+b}{2}+\sqrt{(x-a)(x-b)}\left)\right|+c$