First slide

Methods of integration

Question

The value of $\int_{0}^{π / 2} \frac{d x}{1 + \tan^{3} x}$ is

Easy

A

0
B

1
C

$π / 4$
D

$π / 2$

Solution

$I = \int_{0}^{π / 2} \frac{d x}{1 + \tan^{3} x} = \int_{0}^{π / 2} \frac{d x}{1 + \tan^{3} (π / 2 - x)}$
$\begin{array}{l} = \int_{0}^{π / 2} \frac{d x}{1 + \cot^{3} x} \\ = \int_{0}^{π / 2} \frac{\tan^{3} x}{1 + \tan^{3} x} d x = \int_{0}^{π / 2} (1 - \frac{1}{1 + \tan^{3} x}) d x = \frac{π}{2} - I . \end{array}$
Thus $I = π / 4 .$

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