Question:
$\frac{1}{x \sqrt{a x-x^{2}}}$ [Hint: Put $x=\frac{a}{t}$ ]
Solution:
$\frac{1}{x \sqrt{a x-x^{2}}}$
Let $x=\frac{a}{t} \Rightarrow d x=-\frac{a}{t^{2}} d t$
$\Rightarrow \int \frac{1}{x \sqrt{a x-x^{2}}} d x=\int \frac{1}{\frac{a}{t} \sqrt{a \cdot \frac{a}{t}-\left(\frac{a}{t}\right)^{2}}}\left(-\frac{a}{t^{2}} d t\right)$
$=-\int \frac{1}{a t} \cdot \frac{1}{\sqrt{\frac{1}{t}-\frac{1}{t^{2}}}} d t$
$=-\frac{1}{a} \int \frac{1}{\sqrt{\frac{t^{2}}{t}-\frac{t^{2}}{t^{2}}}} d t$
$=-\frac{1}{a} \int \frac{1}{\sqrt{t-1}} d t$
$=-\frac{1}{a}[2 \sqrt{t-1}]+\mathrm{C}$
$=-\frac{1}{a}\left[2 \sqrt{\frac{a}{x}-1}\right]+\mathrm{C}$
$=-\frac{2}{a}\left(\frac{\sqrt{a-x}}{\sqrt{x}}\right)+\mathrm{C}$
$=-\frac{2}{a}\left(\sqrt{\frac{a-x}{x}}\right)+\mathrm{C}$