First slide

Methods of integration

Question

$\int \frac{1}{1 + \sin x + \cos x} d x$

Moderate

A

$\frac{1}{2} \log |\tan \frac{x}{2} + 1| + C$
B

$\log |\tan (x + 1)| + C$
C

$\log |\tan \frac{x}{2} + 1| + C$
D

None of these

Solution

$Putting \sin x = \frac{2 \tan x / 2}{1 + \tan^{2} x / 2} and \cos x = \frac{1 - \tan^{2} x / 2}{1 + \tan^{2} x / 2},$
$\begin{array}{l} I = \int \frac{1}{1 + \sin x + \cos x} d x \\ = \int \frac{1}{1 + \frac{2 \tan x / 2}{1 + \tan^{2} x / 2} + \frac{1 - \tan^{2} x / 2}{1 + \tan^{2} x / 2}} d x \\ = \int \frac{1 + \tan^{2} x / 2}{1 + \tan^{2} x / 2 + 2 \tan x / 2 + 1 - \tan^{2} x / 2} d x \\ = \int \frac{\sec^{2} x / 2}{2 + 2 \tan x / 2} d x \\ Putting \tan \frac{x}{2} = t and \frac{1}{2} \sec^{2} \frac{x}{2} d x = d t \\ or \sec^{2} \frac{x}{2} d x = 2 d t, we get \\ \begin{array}{l} I = \int \frac{2 d t}{2 + 2 t} = \int \frac{1}{t + 1} d t = \log | t + 1 | + C \\ = \log |\tan \frac{x}{2} + 1| + C \end{array} \end{array}$

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