First slide

Methods of integration

Question

$If I = \int \frac{d x}{{(2 a x + x^{2})}^{3 / 2}}, then I is equal to$

Moderate

A

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

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

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

$- \frac{1}{a^{3}} \frac{x + a}{\sqrt{2 a x + x^{3}}} + C$

Solution

$Write 2 a x + x^{2} = (x + a)^{2} - a^{2}, and put x + a = a \sec θ, so that d x = a \sec θ \tan θ d θ$
$\begin{array}{l} ∴ I = \int \frac{a \sec θ \tan θ}{a^{3} \tan^{3} θ} d θ \\ = \frac{1}{a^{2}} \int \frac{\cos θ}{\sin^{2} θ} d θ = - \frac{1}{a^{2} \sin θ} + C \\ = - \frac{1}{a^{2}} \frac{\sec θ}{\tan θ} + C = - \frac{1}{a^{2}} \frac{x + a}{\sqrt{2 a x + x^{2}}} + C \end{array}$

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