First slide

Introduction to integration

Question

$\int (1 - \cos x) {cosec}^{2} xdx$ is equal to

Easy

A

$\tan x + C$
B

$\tan \frac{x}{2} + C$
C

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

None of these

Solution

$\begin{array}{l} \int (1 - \cos x) {cosec}^{2} xdx \\ \begin{aligned} = \int {cosec}^{2} xdx - \int {cosec}^{2} xcos xdx = \cot x - \int cosec xcot xdx \\ \begin{array}{ll} = & - \cot x + cosec x + C = \frac{1 - \cos x}{\sin x} + C \\ = \frac{2 \sin^{2} \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}} + C \\ = \tan \frac{x}{2} + C \end{array} \end{aligned} \end{array}$

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