Answer
$\pi^{3}-6\pi$
Work Step by Step
We will need
84. $\displaystyle \int u^{n}\sin udu=-u^{n}\cos u+n\int u^{n-1}\cos udu$
85. $\displaystyle \int u^{n}\cos udu=u^{n}\sin u-n\int u^{\mathrm{n}-1}\sin udu$
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$\displaystyle \int x^{3}\sin xdx$=$\qquad$apply 84.
$=-x^{3}\displaystyle \cos x+3\int x^{2}\cos xdx$ =$\qquad$apply 85.
$=-x^{3}\cos x+3\left[x^{2}\sin x-2\int x\sin xdx\right]$
=$\qquad$apply 84
$=-x^{3}\cos x+3\left[x^{2}\sin x-2\left(-x\cos x+\int\cos xdx\right)\right]$
$=-x^{3}\cos x+3\left[x^{2}\sin x-2\left(-x\cos x+\sin x\right)\right]+C$
$\displaystyle \int_{0}^{\pi}x^{3}\sin xdx=\left[-x^{3}\cos x +3x^{2}\sin x+6x\cos x-6\sin x\right]_{0}^{\pi}$
$=(\pi^{3}+0-6\pi-0)-(0+0+0-0)$
$=\pi^{3}-6\pi$
