Question

Evaluate

${\int }_0^a\frac{dx}{\sqrt{ax-x^2}}$

Answer 1

$l{\int }_0^a\frac{dx}{\sqrt{ax-x^2}}$

$I={\int }_0^a\frac{dx}{\sqrt{{\left(\frac{a}{2}\right)}^2-{\left(x-\frac{a}{2}\right)}^2}}$

$Letx-\frac{a}{2}=t$

$dx=dt$

$={\int }_{\frac{-a}{2}}^{\frac{a}{2}}\frac{dt}{\sqrt{{\left(\frac{a}{2}\right)}^2-t^2}}$

$={{\left[{\sin }^{-1}\left(\frac{t}{\frac{a}{2}}\right)\right]}_{\frac{-a}{2}}}^{\frac{a}{2}}$

$=\frac{\pi }{2}-\left(-\frac{\pi }{2}\right)$

Hence solved.