Question

Let \(I=\int_{\pi / 4}^{\pi / 3} \frac{\sin x}{x} d x\). Then
(A) \(\frac{1}{2} \leq 1 \leq 1\)
(B) \(4 \leq 1 \leq 2 \sqrt{30}\)
(C) \(\frac{\sqrt{3}}{8} \leq 1 \leq \frac{\sqrt{2}}{6}\)
(D) \(1 \leq 1 \leq \frac{2 \sqrt{3}}{\sqrt{2}}\)

Answer 1

Ans: (C)
$$
\begin{aligned}
&\text { Hint }: I=\int_{\pi / 4}^{\pi / 3} \frac{\sin x}{x} d x \\
&\Rightarrow \frac{3}{\pi} \times \sin \frac{\pi}{3} \times\left(\frac{\pi}{3}-\frac{\pi}{4}\right) \leq I \leq \frac{4}{\pi} \times \sin \frac{\pi}{4} \times\left(\frac{\pi}{3}-\frac{\pi}{4}\right) \\
&\Rightarrow \frac{\sqrt{3}}{8} \leq I \leq \frac{\sqrt{2}}{6}
\end{aligned}
$$