Question

Integrate:

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

Answer 1

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

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

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

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

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