First slide

Methods of integration

Question

The value of $\int \frac{d x}{5 + 4 \cos x}$ is

Very Easy

A

$\frac{1}{3} \tan^{- 1} (\frac{1}{3} \tan x) + C$
B

$\frac{1}{3} \tan^{- 1} (\frac{1}{3} \tan (\frac{x}{2})) + C$
C

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

$\frac{2}{3} \tan^{- 1} (\frac{1}{3} \tan (\frac{x}{2})) + C$

Solution

$\begin{array}{r} \int \frac{d x}{5 + 4 \cos x} = \int \frac{2 d t}{5 (1 + t^{2}) + 4 (1 - t^{2})} \\ (t = \tan (x / 2)) \end{array}$
$= 2 \int \frac{d t}{9 + t^{2}} = \frac{2}{3} \tan^{- 1} (\frac{1}{3} \tan (\frac{x}{2})) + C$

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