First slide

Evaluation of definite integrals

Question

The value of $\int_{π / 4}^{π / 3} \frac{d x}{\sin x + \tan x}$ is

Moderate

A

3/4
B

1/2
C

$2 / 3 - \sqrt{2} / 2$
D

none of these

Solution

$I = \int_{π / 4}^{π / 3} \frac{\cos x}{\sin x (\cos x + 1)} d x$
$= \int_{π / 4}^{π / 3} \frac{\cos x \sin x}{(1 - \cos^{2} x) (1 + \cos x)} d x$
Put $\cos x = t$ so that
$I = \int_{1 / 2}^{1 / \sqrt{2}} \frac{t d t}{(1 + t)^{2} (1 - t)}$
$= \int_{1 / 2}^{1 / \sqrt{2}} [\frac{1}{4} (\frac{1}{1 + t} + \frac{1}{1 - t}) - \frac{1}{2 (1 + t)^{2}}] d t$
$\begin{array}{l} = {[\frac{1}{4} \log \frac{1 + t}{1 - t} + \frac{1}{2 (1 + t)}]}_{1 / 2}^{1 / \sqrt{2}} \\ = \frac{1}{4} \log [(\frac{\sqrt{2} + 1}{\sqrt{2} - 1}) (\frac{2}{3})] + \frac{1}{2 + \sqrt{2}} - \frac{1}{3} \end{array}$

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