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$\int_0^{\pi / 2} \sin ^8 x \cos ^2 x d x$ is equal to
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$\frac{7 \pi}{512}$
$\begin{aligned} & \int_0^{\pi / 2} \sin ^8 x \cdot \cos ^2 x d x=\frac{\frac{8+1}{2} ! \frac{2+1}{2} !}{2 \frac{8+2+2}{2} !} \\ &=\frac{\frac{7}{2} \cdot \frac{5}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \sqrt{\pi} \cdot \frac{1}{2} \cdot \pi}{2 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}=\frac{7 \pi}{512}\end{aligned}$
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