Question

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

Answer 1

Let $x=\frac{a}{t}\Rightarrow dx=-\frac{a}{t^2}dt$
$\Rightarrow \int \frac{1}{x\sqrt{ax-x^2}}dx=\int \frac{1}{\frac{a}{t}\sqrt{a\cdot \frac{a}{t}-{\left(\frac{a}{t}\right)}^2}}(-\frac{a}{t^2}dt)$
$=-\int \frac{1}{at}\cdot \frac{1}{\sqrt{\frac{1}{t}-\frac{1}{t^2}}}dt$
$=-\frac{1}{a}\int \frac{1}{\sqrt{\frac{t^2}{t}-\frac{t^2}{t^2}}}dt$
$=-\frac{1}{a}\int \frac{1}{\sqrt{t-1}}dt$
$=-\frac{1}{a}\left[2\sqrt{t-1}\right]+C$
$=-\frac{1}{a}\left[2\sqrt{\frac{a}{x}-1}\right]+C$
$=-\frac{2}{a}\left(\frac{\sqrt{a-x}}{\sqrt{x}}\right)+C$
$=-\frac{2}{a}\left(2\sqrt{\frac{a-x}{x}}\right)+C$