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Question
\[\int\limits_0^\pi x \sin x \cos^4 x dx\]
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Solution
\[Let, I = \int_0^\pi x \sin x \cos^4 x d x ............(1)\]
\[ = \int_0^\pi \left( \pi - x \right) \sin\left( \pi - x \right) \cos^4 \left( \pi - x \right) d x \]
\[ = \int_0^\pi \left( \pi - x \right) \sin x \cos^4 x d x ..............(2)\]
Adding (1) and (2)
\[2I = \int_0^\pi \left[ x \sin x \cos^4 x + \left( \pi - x \right) \sin x \cos^4 x \right] d x \]
\[ = \int_0^\pi \left( x + \pi - x \right) \sin x \cos^4 x d x \]
\[ = \pi \int_0^\pi \sin x \cos^4 x d x \]
\[ = \pi \left[ \frac{- \cos^5 x}{5} \right]_0^\pi \]
\[ = \pi\left[ \frac{1}{5} + \frac{1}{5} \right]\]
\[ = \frac{2\pi}{5}\]
\[Hence, I = \frac{\pi}{5}\]
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