Question

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

Answer 1

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

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

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

$=\frac{1}{a}\int \frac{dx}{\sqrt{1-{\left(\frac{x-a}{a}\right)}^2}}$

Let $\frac{x-a}{a}=t\frac{dt}{dx}=\frac{d}{a}$

$dt=\frac{dx}{a}$

$=\int \frac{dt}{\sqrt{1-t^2}}$

$={\sin }^{-1}t+c$

$={\sin }^{-1}\left(\frac{x-a}{a}\right)+c$