First slide

Methods of integration

Question

The primitive of $\frac{1}{(x - a)^{3 / 2} (b - x)^{1 / 2}}$ is

Moderate

A

$\frac{1}{b - a} {[\frac{b - x}{x - a}]}^{1 / 2} + C$
B

$\frac{3}{4 (b - a)} {[\frac{b - x}{x - a}]}^{1 / 2} + C$
C

$\frac{1}{b - a} {[\frac{x - a}{b - a}]}^{1 / 2} + C$
D

none of these

Solution

putting $x - a = (1 / t)$ we have $d x = (- 1 / t^{2}) d t$
$and the given integral can be written as$
$\int \frac{- \frac{1}{t^{2}} d t}{(1 / t)^{3 / 2} {(b - a - \frac{1}{t})}^{1 / 2}} = - \int \frac{d t}{\sqrt{(b - a) t - 1}}$
$\begin{array}{l} = - 2 \frac{\sqrt{(} b - a) t - 1}{b - a} + C \\ = \frac{- 2}{b - a} \sqrt{\frac{b - a - (x - a)}{x - a}} + C \\ = \frac{2}{a - b} \sqrt{\frac{b - x}{x - a}} + C \end{array}$

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