Integrate the function: $\frac{1}{x \sqrt{ax - x^{2}}} \quad \left[ \text{Hint: } x = \frac{a}{t} \right]$

Let $I = \int \frac{1}{x \sqrt{ax - x^{2}}} dx$.
Substitute $x = \frac{a}{t}$,then $dx = -\frac{a}{t^{2}} dt$.
Substituting these into the integral:
$I = \int \frac{1}{\frac{a}{t} \sqrt{a \cdot \frac{a}{t} - \left(\frac{a}{t}\right)^{2}}} \left(-\frac{a}{t^{2}} dt\right)$
$I = \int \frac{1}{\frac{a}{t} \sqrt{\frac{a^{2}}{t} - \frac{a^{2}}{t^{2}}}} \left(-\frac{a}{t^{2}} dt\right)$
$I = \int \frac{1}{\frac{a}{t} \cdot \frac{a}{t} \sqrt{t - 1}} \left(-\frac{a}{t^{2}} dt\right)$
$I = -\int \frac{1}{\sqrt{t - 1}} dt$
$I = -2 \sqrt{t - 1} + C$
Substituting $t = \frac{a}{x}$ back:
$I = -2 \sqrt{\frac{a}{x} - 1} + C = -2 \sqrt{\frac{a - x}{x}} + C$.