First slide

Evaluation of definite integrals

Question

The value of $\int_{0}^{α} \frac{d x}{1 - \cos α \cos x} (0 < α < π / 2)$ is

Moderate

A

$\frac{π}{\sin α}$
B

$\frac{π}{2 \cos α}$
C

$\frac{π}{\cos α}$
D

$\frac{π}{2 \sin α}$

Solution

Putting $\tan \frac{x}{2} = t$ the given integral reduces to
$\begin{array}{l} = 2 \int_{0}^{\tan \frac{α}{2}} \frac{d t}{(1 - \cos α) + t^{2} (1 + \cos α)} \\ = \frac{1}{\cos^{2} \frac{α}{2}} \int_{0}^{\tan α / 2} \frac{d t}{t^{2} + \tan^{2} \frac{α}{2}} \\ = \frac{1}{{\cos^{2} \frac{α}{2} \tan \frac{α}{2} \tan^{- 1} \frac{t}{\tan \frac{α}{2}}|}_{0}^{\tan α / 2}} \\ = \frac{1}{\sin \frac{α}{2} \cos \frac{α}{2}} \cdot \frac{π}{4} = \frac{π}{2 \sin α} \end{array}$

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