First slide

Evaluation of definite integrals

Question

$\int_{- π}^{π} \frac{2 x (1 + \sin x)}{1 + \cos^{2} x} d x =$

Moderate

A

$\frac{π^{2}}{4}$
B

$π^{2}$
C

0
D

$\frac{π}{2}$

Solution

$I = \int_{- π}^{π} \frac{2 x \sin x d x}{1 + \cos^{2} x}$ dropping odd term
$I = 4 \int_{0}^{π} \frac{x \sin x}{1 + \cos^{2} x} d x, x \to π - x$

$I = 4 \int_{0}^{π} \frac{(π - x) \sin x}{1 + \cos^{2} x} d x$

$2 I = 4 π \int_{0}^{π} \frac{\sin x}{1 + \cos^{2} x} d x = 8 π \int_{0}^{\frac{π}{2}} \frac{\sin x}{1 + \cos^{2} x} d x$

$= 8 π \tan^{- 1} {(- \cos x) |}_{0}^{\frac{π}{2}} = 8 π (0 + \frac{π}{4})$

$= 2 π^{2} \to I = π^{2}$

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