Sine-weighted power integral recurrence ($F(k)={\pi }^k-k(k-1)F(k-2)$)

Define the definite integral $F(k)={\int }_0^{\pi }{x}^k\sin xdx$. For real $k>1$, two applications of integration by parts giveThe boundary term $x^{k-1}\sin x$ vanishes at both endpoints. At $k=1$, direct integration gives $F(1)=\pi$, consistent with the displayed relation; $F(-1)$ is finite because $\sin x/x\to 1$ at zero. This recurrence and $F(0)=2$ evaluate every nonnegative integer moment, with separate even and odd chains.