First slide

Methods of integration

Question

$\int \frac{d x}{{(a^{2} - b^{2} x^{2})}^{3 / 2}}$ equals

Easy

A

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

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

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

$\frac{1}{a^{2} \sqrt{a^{2} - b^{2} x^{2}}} + C$

Solution

Put $x = \frac{a}{b} \sin θ \Rightarrow d x = \frac{a}{b} \cos θ d θ$
$\begin{array}{l} \int \frac{d x}{{(a^{2} - b^{2} x^{2})}^{3 / 2}} = \frac{a}{b} \int \frac{\cos θ d θ}{a^{3} \cos^{3} θ} = \frac{1}{a^{2} b} \tan θ + C \\ = \frac{1}{a^{2} b} \frac{\sin θ}{\sqrt{1 - \sin^{2} θ}} + C \\ = \frac{1}{a^{2} b} \frac{\frac{b x}{a}}{\sqrt{1 - \frac{b^{2}}{a^{2}} x^{2}}} + C = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} - b^{2} x^{2}}} + C \end{array}$

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