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Question
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
Solution
\[\int_0^\frac{\pi}{4} \cos^4 x \sin^3 x d x\]
\[ = \int_0^\frac{\pi}{4} \cos^4 x \sin x \left( 1 - \cos^2 x \right) dx\]
\[ = \int_0^\frac{\pi}{4} \cos^4 x \sin x dx - \int_0^\frac{\pi}{4} \cos^6 x \sin x dx\]
\[ = - \left[ \frac{\cos^5 x}{5} \right]_0^\frac{\pi}{4} + \left[ \frac{\cos^7 x}{7} \right]_0^\frac{\pi}{4} \]
\[ = \frac{- 1}{20\sqrt{2}} + \frac{1}{5} + \frac{1}{56\sqrt{2}} - \frac{1}{7}\]
\[ = \frac{- \sqrt{2}}{40} + \frac{2}{35} + \frac{\sqrt{2}}{112}\]
\[ = \frac{2}{35} - \frac{9\sqrt{2}}{560}\]
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