First slide

Evaluation of definite integrals

Question

$\int_{0}^{π / 2} \frac{d x}{2 \cos x + 3}$ is equal to

Easy

A

$\frac{2}{\sqrt{5}} \tan^{- 1} \frac{1}{\sqrt{5}}$
B

$\frac{1}{\sqrt{5}} \tan^{- 1} \frac{1}{\sqrt{5}}$
C

1
D

$\frac{2}{\sqrt{3}} \tan^{- 1} \frac{1}{\sqrt{5}}$

Solution

Put $\tan x / 2 = t$ The given integral reduces to
$\begin{matrix} \int_{0}^{1} \frac{2 d t}{(1 + t^{2}) (2 \frac{(1 - t^{2})}{1 + t^{2}} + 3)} = 2 \int_{0}^{1} \frac{d t}{2 - 2 t^{2} + 3 + 3 t^{2}} \\ = 2 \int_{0}^{1} \frac{d t}{t^{2} + 5} \\ = {\frac{2}{\sqrt{5}} \tan^{- 1} \frac{t}{\sqrt{5}}|}_{0}^{1} = \frac{2}{\sqrt{5}} \tan^{- 1} \frac{1}{\sqrt{5}} \end{matrix}$

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