First slide

Methods of integration

Question

$\int \frac{dx}{2 + \sin x + \cos x}$ is equal to

Easy

A

$\sqrt{2} \tan^{- 1} (\frac{\tan x / 2 + 1}{\sqrt{2}}) + C$
B

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

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

None of these

Solution

Let $I = \int \frac{dx}{2 + \sin x + \cos x}$
$\begin{array}{l} \Rightarrow I = \int \frac{dx}{2 + \frac{2 \tan \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}} + \frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}}} \\ = \int \frac{\sec^{2} \frac{x}{2} dx}{2 + 2 \tan^{2} \frac{x}{2} + 2 \tan \frac{x}{2} + 1 - \tan^{2} \frac{x}{2}} \\ I = \int \frac{\sec^{2} \frac{x}{2} dx}{\tan^{2} \frac{x}{2} + 2 \tan \frac{x}{2} + 3} \end{array}$
Put $\tan \frac{x}{2} = t \Rightarrow \frac{1}{2} \sec^{2} \frac{x}{2} dx = dt$
$\begin{array}{l} ∴ I = \int \frac{2 dt}{t^{2} + 2 t + 3} = 2 \int \frac{dt}{t^{2} + 2 t + 1 + 2} \\ = 2 \int \frac{dt}{(t + 1)^{2} + (\sqrt{2})^{2}} = 2 \cdot \frac{1}{\sqrt{2}} \tan^{- 1} (\frac{t + 1}{\sqrt{2}}) + C \\ \Rightarrow I = \sqrt{2} \tan^{- 1} (\frac{\tan x / 2 + 1}{\sqrt{2}}) + C \end{array}$

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