Question

Let $I=\underset{\pi /4}{\overset{\pi /3}{\int }}\frac{\sin x}{x}dx$ .Then

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

Answer 1

Answer: (C)

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