SATHEE: Indefinite Integration 1 Question 26

Indefinite Integration 1 Question 26

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

(2001, 1M)

(a) $π$

(b) $a π$

(d) $2 π$

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Solution:

Let $I = \int_{- π}^{π} \frac{\cos^{2} x}{1 + a^{x}} d x$

$\begin{aligned} = \int_{π}^{- π} \frac{\cos^{2} (- x)}{1 + a^{- x}} d (- x) \\ \Rightarrow I & = \int_{- π}^{π} a^{x} \frac{\cos^{2} x}{1 + a^{x}} d x \end{aligned}$

On adding Eqs. (i) and (ii), we get

$\begin{aligned} 2 I & = \int_{- π}^{π} \frac{1 + a^{x}}{1 + a^{x}} \cos^{2} x d x \\ = \int_{- π}^{π} \cos^{2} x d x = 2 \int_{0}^{π} \frac{1 + \cos 2 x}{2} d x \\ = \int_{0}^{π} (1 + \cos 2 x) d x = \int_{0}^{π} 1 d x + \int_{0}^{π} \cos 2 x d x \\ = [x]_{0}^{π} + 2 \int_{0}^{π / 2} \cos 2 x d x = π + 0 \\ \Rightarrow 2 I & = π \Rightarrow I = π / 2 \end{aligned}$

$e^{\cos x} \sin x, for | x | \leq 2$

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Indefinite Integration 1 Question 26

27. The value of ∫−ππcos2⁡x1+axdx,a>0, is

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