Question

Integrate $\int \frac{\sqrt{a-x}}{x}dx$

Answer 1

Let $x=a{\sin }^2\theta$

$\Rightarrow dx=2a\sin \theta \cos \theta d\theta$

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

$=2\sqrt{a}\int \frac{1-{\sin }^2\theta }{\sin \theta }d\theta =2\sqrt{a}\left[\frac{1}{\sin \theta }d\theta -\int \sin \theta d\theta \right]$

$=2\sqrt{a}\left[\int cosec\theta d\theta -\int \sin \theta d\theta \right]$

$=2\sqrt{a}[ln|cosec\theta -\cot \theta |+\cos \theta ]+c$

${\sin }^2\theta =\frac{x}{a}$

$\Rightarrow \sin \theta =\sqrt{\frac{x}{a}},\cos \theta =\sqrt{1-\frac{x}{a}}$

$\Rightarrow I=2\sqrt{a}[ln|\sqrt{\frac{a}{x}}-\frac{\sqrt{a-x}}{\sqrt{x}}|+\sqrt{1-\frac{x}{a}}]+c$.