Question

Integrate the following functions w.r.t. x.

$\int \frac{1}{x\sqrt{ax-x^2}}dx$

Answer 1

Let I = $\int \frac{1}{x\sqrt{ax-x^2}}dxPutx=\frac{a}{t}\Rightarrow dx=\frac{-adt}{t^2}\therefore I=\int \frac{1}{\frac{a}{t}\sqrt{a\left(\frac{a}{t}\right)-\frac{a^2}{t^2}}}\left(\frac{-a}{t^2}\right)dt=\int \frac{-a}{a.a\sqrt{t-1}}dt=-\frac{1}{a}\int (t-1{)}^{-1/2}dt=-\frac{1}{a}.\frac{(t-1{)}^{(-1/2)+1}}{-\left(1/2\right)+1}+C=-\frac{2}{a}\sqrt{t-1}+C=-\frac{2}{a}\sqrt{\frac{a}{x}-1}=C(\because x=\frac{a}{t}\Rightarrow t=\frac{a}{x})=-\frac{2}{a}\sqrt{\frac{a-x}{x}}+C$