Question

Integrate the function: $\frac{1}{x\sqrt{ax-x^2}}$

Answer 1

Let $I=\int \frac{1}{x\sqrt{ax-x^2}}dx$
$\Rightarrow I=\int \frac{1}{x^2\sqrt{\frac{a}{x}-1}}dx$ ...(1)
Substituing $\frac{a}{x}-1=t^2$
Differentiating w.r.t. $x$
$-\frac{a}{x^2}=2t\frac{dt}{dx}\Rightarrow \frac{dx}{x^2}=-2t\frac{dt}{a}$
So,
$I=\int \frac{1}{x^2\sqrt{\frac{a}{x}-1}}dx$
$\Rightarrow I=-\frac{1}{a}\int \frac{2t}{t}dt$
$\Rightarrow I=-\frac{2}{a}\int 1dt$
$\Rightarrow I=-\frac{2}{a}t+C$
$\Rightarrow I=-\frac{2}{a}\sqrt{\frac{a}{x}-1+C}$
Hence, $\int \frac{1}{x\sqrt{ax-x^2}}dx=-\frac{2}{a}\sqrt{\frac{a-x}{x}}+C$
Where $C$ is constant of integration.
Where $C$ is constant of integration.