題目

Problem

Suppose that $c$ is a positive constant. What is $\displaystyle \int_0^{2\pi/c} x\sin(cx)\,dx$ ?

(a) $2\pi/c$

(b) $\pi c$

(d) $-\pi c$

(e) None of the above.

解答

展開

展開

\begin{align*} \int_0^{2\pi/c}x\sin(cx)\,\mathrm{d}x =&\,\left[-\frac{x\cos(cx)}{c}+\frac{\sin(cx)}{c^2}\right]_0^{2\pi/c}\\[4mm] =&\,-\frac{2\pi}{c^2}. \end{align*}

此值不在 (a)–(d)，答案為 $\boxed{\text{(e)}}$ 。