First slide

Evaluation of definite integrals

Question

$\int_{π / 4}^{3 π / 4} \frac{d x}{1 + \cos x}$ is equal to

Easy

A

2
B

– 2
C

1/2
D

– 1/2

Solution

$I = \int_{π / 4}^{3 π / 4} \frac{d x}{1 + \cos x} = \int_{π / 4}^{3 π / 4} \frac{d x}{1 + \cos (π - x)}$
$\begin{array}{l} = \int_{π / 4}^{3 π / 4} \frac{d x}{1 - \cos x} \\ ∴ 2 I = \int_{π / 4}^{3 π / 4} (\frac{1}{1 + \cos x} + \frac{1}{1 - \cos x}) d x \\ = \int_{π / 4}^{3 π / 4} \frac{2}{1 - \cos^{2} x} d x \\ = - {2 \cot x|}_{π / 4}^{3 π / 4} = 4 . Hence I = 2 \end{array}$

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