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$\frac{1}{x \sqrt{a x-x^{2}}}$

Let $x=\frac{a}{t} \Rightarrow d x=-\frac{a}{t^{2}} d t$

$\Rightarrow \int \frac{1}{x \sqrt{a x-x^{2}}} d x=\int \frac{1}{\frac{a}{t} \sqrt{a \cdot \frac{a}{t}-\left(\frac{a}{t}\right)^{2}}}\left(-\frac{a}{t^{2}} d t\right)$

$=-\int \frac{1}{a t} \cdot \frac{1}{\sqrt{\frac{1}{t}-\frac{1}{t^{2}}}} d t$

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

$=-\frac{1}{a} \int \frac{1}{\sqrt{t-1}} d t$

$=-\frac{1}{a}[2 \sqrt{t-1}]+\mathrm{C}$

$=-\frac{1}{a}\left[2 \sqrt{\frac{a}{x}-1}\right]+\mathrm{C}$

$=-\frac{2}{a}\left(\frac{\sqrt{a-x}}{\sqrt{x}}\right)+\mathrm{C}$

$=-\frac{2}{a}\left(\sqrt{\frac{a-x}{x}}\right)+\mathrm{C}$