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Question: Let I<sub>1</sub> = \(\int_{\pi/6}^{\pi/3}\frac{\sin x}{x}\)dx, I<sub>2</sub> = \(\int_{\pi/6}^{\pi/...

Let I1 = ∫π/6π/3sin⁡xx\int_{\pi/6}^{\pi/3}\frac{\sin x}{x}dx, I2 = ∫π/6π/3sin⁡(sin⁡x)sin⁡x\int_{\pi/6}^{\pi/3}\frac{\sin(\sin x)}{\sin x}dx,

I3 = ∫π/6π/3sin⁡(tan⁡x)tan⁡x\int_{\pi/6}^{\pi/3}\frac{\sin(\tan x)}{\tan x}dx, then

A

I1< I2< I3

B

I2< I1< I3

C

I3< I1< I2

D

I3< I2< I1

Answer

I3< I1< I2

Explanation

Solution

f(x) = sin⁡xx\frac{\sin x}{x}is a decreasing function and

sin⁡xx\frac{\sin x}{x} > 0 for all x in (0, p)

Since sin x < x < tan x

Ž sin⁡(sin⁡x)sin⁡x\frac{\sin(\sin x)}{\sin x} > sin⁡xx\frac{\sin x}{x} > sin⁡(tan⁡x)tan⁡x\frac{\sin(\tan x)}{\tan x} for π6\frac{\pi}{6} < x < π3\frac{\pi}{3}

\ I2 > I1 > I3