Question

$\underset{a}{\overset{b}{\int }}\frac{1}{\sqrt{\left(x-a\right)\left(b-x\right)}}dx,d>a$

Answer 1

$\underset{a}{\overset{b}{\int }}\frac{dx}{\sqrt{\left(x-a\right)\left(b-x\right)}}$
when $x=a,c=a{\sin }^2\theta +b{\cos }^2\theta$
$\Rightarrow a{\cos }^2\theta =b{\cos }^2\theta \Rightarrow \cos \theta =\pi /2$
when $x,=b,$
$b=a{\sin }^2\theta +b{\cos }^2\theta =0$
$\underset{\pi /2}{\overset{0}{\int }}\frac{2\sin \theta \cos \theta \left(a-b\right)}{\sqrt{\left(b-a\right){\cos }^2\theta \sqrt{\left(b-a\right){\sin }^2\theta }}}$
$\underset{\pi /2}{\overset{0}{\int }}\frac{2\sin \theta \cos \theta \left(a-b\right)}{\left(b-a\right)\sin \theta \cos \theta }$
$=\left(a-b\right)\times 2\times \pi /2$
$=\pi \left(b-a\right)$