Mathematics

$$\underset{0}{\overset{\pi}{\int}} x \, f (\sin \, x) dx = $$


ANSWER

$$\pi \underset{0}{\overset{\pi}{\int}} f (\sin \, x) dx$$


SOLUTION
$$\displaystyle\int_{0}^{\pi } xf(\sin x)dx$$

$$=\displaystyle\int_{0}^{\pi } f(\sin x)dx\left [ \int_{0}^{\pi }x  \sin x dx \right ]$$

$$=\displaystyle\int_{0}^{\pi } f(\sin x)dx\left [ -x \cos x -\int_{0}^{\pi}-\cos x dx \right ]$$

$$=\displaystyle\int_{0}^{\pi } f(\sin x)dx\left [ -x \cos x +sinx \right ]_0^{\pi}$$

$$=\displaystyle\int_{0}^{\pi } f(\sin x)dx(\pi-0)$$

$$=\displaystyle\pi \int_{0}^{\pi } f(\sin x)dx$$
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Single Correct Medium Published on 17th 09, 2020
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