Question

Let $I = \int\limits^{\pi /3}_{\pi / 4} \frac{\sin \, x}{x} dx$ .Then

• A. $\frac{1}{2} \leq I \leq 1$
• B. $4 \leq I \leq 2 \sqrt{30}$
• C. $\frac{\sqrt{3}}{8} \leq I \leq \frac{\sqrt{2}}{6}$
• D. $1 \leq I \leq \frac{2 \sqrt{3}}{\sqrt{2}}$

Answer 1

Answer: (C)

We have,
$I = \int\limits_{\pi/4}^{\pi/4} \frac{\sin \,x}{x} dx$
Since, $\frac{\sin x}{x}$ is a decreasing function.
$\therefore \frac{\pi}{12} \times \frac{\sin \pi / 3}{\pi / 3} \leq 1 \leq \frac{\pi}{12} \times \frac{\sin \pi / 4}{\pi / 4}$
$\Rightarrow \frac{\sqrt{3}}{8} \leq I \leq \frac{\sqrt{2}}{6}$