First slide

Methods of integration

Question

$If J = \int \frac{1}{\tan^{2} x + \sec^{2} x} d x then J is equal to$

Moderate

A

$\sqrt{2} \tan^{- 1} (\sqrt{2} \tan x) - \tan^{- 1} (\sqrt{2} \tan x) + K$
B

$\sqrt{2} \tan^{- 1} (\sqrt{2} \tan x) - x + K$
C

$\sqrt{2} \tan^{- 1} (\sqrt{2} \tan x) + \frac{x}{2} + K$
D

$\sqrt{2} \tan^{- 1} (\sqrt{2} \tan x) + x + K$

Solution

$Let J = \int \frac{1}{\tan^{2} x + \sec^{2} x} d x$
$Let J = \int \frac{1}{(\tan^{2} x + \sec^{2} x)} \frac{\sec^{2} x}{\sec^{2} x} d x$
$= \int \frac{\sec^{2} x d x}{(2 \tan^{2} x + 1) (1 + \tan^{2} x)}$
$Put \tan x = t \Rightarrow \sec^{2} x d x = d t$
$Let J = \int \frac{d t}{(2 t^{2} + 1) (1 + t^{2})} = \int [\frac{2}{2 t^{2} + 1} - \frac{1}{1 + t^{2}}] d t$
$\begin{array}{l} = \frac{2}{\sqrt{2}} \tan^{- 1} (\sqrt{2} t) - \tan^{- 1} (t) + K \\ J = \sqrt{2} \tan^{- 1} (\sqrt{2} \tan x) - \tan^{- 1} (\tan x) + K \\ J = \sqrt{2} \tan^{- 1} (\sqrt{2} \tan x) - x + K \end{array}$

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