First slide

Evaluation of definite integrals

Question

The value of $\int_{- π}^{π} \frac{\cos^{2} x}{1 + a^{x}} d x, a > 0$ is

Easy

A

$π / 2$
B

$a π$
C

$π$
D

$2$ $π$

Solution

Let $I = \int_{- π}^{π} \frac{\cos^{2} x}{1 + a^{x}} d x$
Put $x = - θ$ to obtain
$I = \int_{π}^{- π} \frac{\cos^{2} θ}{1 + a^{- θ}} (- 1) d θ = \int_{- π}^{π} \frac{a^{x} \cos^{2} x}{1 + a^{x}} d x$
Adding (1) and (2), we get
$\begin{array}{l} 2 I = \int_{- π}^{π} \cos^{2} x d x = 2 \int_{0}^{π} \cos^{2} x d x = π \\ \Rightarrow I = π / 2 \end{array}$

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