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Question: Let I<sub>1</sub> = \(\int_{a}^{\pi –a}{xf(\sin x)dx,}\) I<sub>2</sub> = \(\int_{a}^{\pi –a}{f(\sin ...

Let I1 = ∫aπ–axf(sin⁡x)dx,\int_{a}^{\pi –a}{xf(\sin x)dx,} I2 = ∫aπ–af(sin⁡x)dx\int_{a}^{\pi –a}{f(\sin x)dx}, then I2 is equal to -

A

π2\frac{\pi}{2} I1

B

pI1

C

2π\frac{2}{\pi} I1

D

2I1

Answer

2π\frac{2}{\pi} I1

Explanation

Solution

I­1=∫aπ–axf(sin⁡x)dx\int_{a}^{\pi –a}{xf(\sin x)dx} A+B–x=a+(p–a)–x=p–x

f(p–x)=f[sin(p–x)]=f(sin x)\I­1=p–a+a/2∫aπ–af(sin⁡x)dx\int_{a}^{\pi –a}{f(\sin x)dx}

ŽI1 = p/2 I2 Ž I1/I2 = p/2