First slide

Evaluation of definite integrals

Question

Let $I = \int_{0}^{π / 2} \frac{d x}{5 + 3 \cos^{2} x}$
Statement-1: $\frac{π}{16} \leq I \leq \frac{π}{10}$
Statement-2: $\frac{1}{8} \leq \frac{1}{5 + 3 \cos^{2} x} \leq \frac{1}{5}$

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

Since $0 \leq \cos^{2} x \leq 1$ so
$\frac{1}{8} \leq \frac{1}{5 + 3 \cos^{2} x} \leq \frac{1}{5}$
$\Rightarrow \frac{π}{16} \leq \int_{0}^{π / 2} \frac{d x}{5 + 3 \cos^{2} x} \leq \frac{π}{10}$

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