First slide

Methods of integration

Question

The value of $\int {(x + \frac{1}{x})}^{3 / 2} (\frac{x^{2} - 1}{x^{2}}) d x,$ is

Moderate

A

$\frac{2}{3} {(x + \frac{1}{x})}^{3 / 2} + C$
B

$\frac{2}{5} {(x + \frac{1}{x})}^{5 / 2} + C$
C

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

none of these

Solution

We have,
$\begin{array}{l} I = \int {(x + \frac{1}{x})}^{3 / 2} (\frac{x^{2} - 1}{x^{2}}) d x \\ \Rightarrow I = \int {(x + \frac{1}{x})}^{3 / 2} d (x + \frac{1}{x}) = \frac{2}{5} {(x + \frac{1}{x})}^{5 / 2} + C \end{array}$

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