First slide

Evaluation of definite integrals

Question

$Let I_{1} = \int_{π / 6}^{π / 3} \frac{\sin x}{x} d x, I_{2} = \int_{π / 6}^{π / 3} \frac{\sin (\sin x)}{\sin x} d x and I_{3} = \int_{π / 6}^{π / 3} \frac{\sin (\tan x)}{\tan x} d x T hen arrange I_{1}, I_{2} and I_{3} in the decreasing order of their values.$

NA

A

$I_{1} > I_{2} > I_{3}$
B

$I_{2} > I_{1} > I_{3}$
C

$I_{3} > I_{2} > I_{1}$
D

$I_{1} = I_{2} = I_{3}$

Solution

$f (x) = \frac{\sin x}{x} is a decreasing function and \frac{\sin x}{x} > 0$ $for all x in (0, π) .$
$\begin{aligned} Since \sin x < x < \tan x \\ \frac{\sin (\sin x)}{\sin x} > \frac{\sin x}{x} > \frac{\sin (\tan x)}{\tan x} for \frac{π}{6} < x < \frac{π}{3} \end{aligned}$
$∴ I_{2} > I_{1} > I_{3}$

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