Let $$I1=$$∫π$$/6$$π$$/3$$ $$sin$$⁡xxdx,$$I2=$$∫π$$/6$$π$$/3$$ $$sin$$⁡$$($$$$sin$$⁡$$x)$$$$sin$$⁡xdx and $$I3=$$∫π$$/6$$π$$/3$$ $$sin$$⁡$$($$$$tan$$⁡$$x)$$$$tan$$⁡xdx Then arrange $$I1$$,$$I2$$ and $$I3$$ in the decreasing order of their values.

Question

Let $$I1=$$∫π$$/6$$π$$/3$$ $$sin$$⁡xxdx,$$I2=$$∫π$$/6$$π$$/3$$ $$sin$$⁡$$($$$$sin$$⁡$$x)$$$$sin$$⁡xdx and $$I3=$$∫π$$/6$$π$$/3$$ $$sin$$⁡$$($$$$tan$$⁡$$x)$$$$tan$$⁡xdx  Then arrange $$I1$$,$$I2$$ and $$I3$$ in the  decreasing order of their values.

Infinity Learn · Accepted Answer

$$f(x)=$$$$sin$$⁡xx is a decreasing function and $$sin$$⁡xx>$$0$$ for all x in (0$$,π$$) .  Since $$sin$$⁡xsin⁡xx>$$sin$$⁡$$($$$$tan$$⁡$$x)$$$$tan$$⁡x for π$$6I1$$>$$I3$$

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

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Let I1=∫π/6π/3 sin⁡xxdx,I2=∫π/6π/3 sin⁡(sin⁡x)sin⁡xdx and I3=∫π/6π/3 sin⁡(tan⁡x)tan⁡xdx Then arrange I1,I2 and I3 in the decreasing order of their values.

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