First slide

Methods of integration

Question

$\int \frac{dx}{x \sqrt{ax - x^{2}}}$ is equal to

Moderate

A

$- \frac{1}{a} \sqrt{\frac{x - a}{x}} + C$
B

$\frac{- 2}{a} \sqrt{\frac{a - x}{x}} + C$
C

$\frac{2}{a} \sqrt{\frac{a - x}{x}} + C$
D

None of these

Solution

Let
$I = \int \frac{dx}{x \sqrt{ax - x^{2}}}, put x = \frac{a}{t} \Rightarrow dx = - \frac{adt}{t^{2}}$
$\begin{array}{l} ∴ l = \int \frac{1}{\frac{a}{t} \sqrt{a (\frac{a}{t}) - \frac{a^{2}}{t^{2}}}} \cdot (\frac{- a}{t^{2}}) dt = \int \frac{- a}{a \cdot a \sqrt{t - 1}} \cdot dt \\ = - \frac{1}{a} \int (t - 1)^{- 1 / 2} \cdot dt = \frac{- 1}{a} \cdot \frac{(t - 1)^{(- 1 / 2) + 1}}{- (1 / 2) + 1} + C \\ = - \frac{2}{a} \sqrt{t - 1} + C = \frac{- 2}{a} \sqrt{\frac{a}{x} - 1} + C [∵ x = \frac{a}{t} \Rightarrow t = \frac{a}{x}] \\ = \frac{- 2}{a} \sqrt{\frac{a - x}{x}} + C \end{array}$

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