First slide

Evaluation of definite integrals

Question

The value of the integral $\int_{- π / 3}^{π / 3} \frac{x \sin x}{\cos^{2} x} d x$ is

Moderate

A

$(\frac{π}{3} - \log \tan \frac{3 π}{2})$
B

$2 (\frac{2 π}{3} - \log (2 + \sqrt{3}))$
C

$3 (\frac{π}{2} - \log \tan \frac{π}{12})$
D

none of these

Solution

The integrand is an even function, so the given integral is equal to
$\begin{array}{l} 2 \int_{0}^{π / 3} \frac{x \sin x}{\cos^{2} x} d x = 2 [x \sec x]_{0}^{π} - \int_{0}^{π / 3} \sec x d x] \\ = 2 \cdot \frac{π}{3} \cdot 2 - 2 [\log (\sec x + \tan x)]_{0}^{5} \\ = \frac{4 π}{3} - 2 \log (2 + \sqrt{3}) \end{array}$

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