Tích phân \(I = \int\limits_{\frac{\pi }{3}}^{\frac{\pi }{2}} {x\sin ax} dx,{\rm{ }}a \ne 0\) có giá trị l:

* Hướng dẫn giải

Đặt \(\left\{ \begin{array}{l} u = x\\ dv = \sin axdx \end{array} \right. \Rightarrow \left\{ \begin{array}{l} du = dx\\ v = - \frac{1}{a}\cos x \end{array} \right.\).

\(\begin{array}{l} \Rightarrow I = \left. {\left( {\frac{{ - 1}}{a}x\cos x} \right)} \right|_{\frac{\pi }{3}}^{\frac{\pi }{2}} + \frac{1}{a}\int\limits_{\frac{\pi }{3}}^{\frac{\pi }{2}} {\cos xdx} \\ = \left. {\left( {\frac{{ - 1}}{a}x\cos x} \right)} \right|_{\frac{\pi }{3}}^{\frac{\pi }{2}} + \left. {\left( {\frac{1}{a}\sin x} \right)} \right|_{\frac{\pi }{3}}^{\frac{\pi }{2}}\\ = \frac{{\pi + 6 - 3\sqrt 3 }}{{6a}} \end{array}\).