First slide

Methods of integration

Question

$\int \sec^{4} x {cosec}^{2} x d x, is equal to$

Moderate

A

$3 \tan^{3} x + 2 \tan x - \tan x + C$
B

$\frac{1}{3} \tan^{3} x + \tan x - \cot x + C$
C

$\frac{1}{3} \tan^{3} x + 2 \tan x - \cot x + C$
D

None of these

Solution

We have
$\begin{array}{l} I = \int \sec^{4} x {cosec}^{2} x d x = \int (1 + \tan^{2} x) \sec^{2} x (1 + \cot^{2} x) d x \\ = \int (1 + t^{2}) (1 + \frac{1}{t^{2}}) d t [putting \tan x = t] \\ = \int (t^{2} + 2 + \frac{1}{t^{2}}) d t = \frac{1}{3} t^{3} + 2 t - \frac{1}{t} + C \\ = \frac{1}{3} \tan^{3} x + 2 \tan x - \cot x + C \end{array}$

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