Ta có:
$\int\limits_0^{\pi}\cos^3x\sin8xdx$
$=\int\limits_0^{\pi}\dfrac{(3\cos x+\cos3x)\sin8x}{4}dx$
$=\int\limits_0^{\pi}\left[\dfrac{3}{8}(\sin9x+\sin7x)+\dfrac{1}{8}(\sin11x+\sin5x)\right]dx$
$=\left(-\dfrac{\cos9x}{24}-\dfrac{3\cos7x}{56}-\dfrac{\cos5x}{40}-\dfrac{\cos11x}{88}\right)\left|\begin{array}{l}\pi\\0\end{array}\right.$
$=\dfrac{304}{1155}$