First slide

Methods of integration

Question

$\int \frac{dx}{{(a^{2} + x^{2})}^{3 / 2}}$ is equal to

Easy

A

$\frac{x}{{(a^{2} + x^{2})}^{1 / 2}} + C$
B

$\frac{x}{a^{2} {(a^{2} + x^{2})}^{1 / 2}} + C$
C

$\frac{1}{a^{2} {(a^{2} + x^{2})}^{1 / 2}} + C$
D

None of these

Solution

$I = \int \frac{dx}{{(a^{2} + x^{2})}^{3 / 2}}$
put , $x = atan θ \Rightarrow dx = {asec}^{2} θdθ$
$\begin{array}{l} ∴ I = \int \frac{{asec}^{2} θ}{{(a^{2} + a^{2} \tan^{2} θ)}^{3 / 2}} dθ \\ = \int \frac{{asec}^{2} θ}{a^{3} {(\sec^{2} θ)}^{3 / 2}} dθ \\ \Rightarrow I = \frac{1}{a^{2}} \int \frac{dθ}{\sec θ} = \frac{1}{a^{2}} \int \cos θdθ = \frac{1}{a^{2}} \sin θ + C \\ \Rightarrow = \frac{x}{a^{2} {(x^{2} + a^{2})}^{1 / 2}} + C \end{array}$

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