$x^3$ $3x^2$ $6x$ $6$ $0$
$\sin x$ $-\cos x$ $-\sin x$ $\cos x$ $\sin x$
$符号$ $+$ $-$ $+$ $-$

\begin{aligned} \int x^3\sin x dx &= +(x^3)(-\cos x)-(3x^2)(-\sin x)+(6x\cos x)-(6\sin x) \\ &= -x^3\cos x+3x^2\sin x + 6x\cos x - 6\sin x + c \end{aligned}