Tính tích phân $I = \int\limits_0^{\frac{\pi }{2}} {x\cos \left( {a - x} \right)dx}$

Đặt $\left\{ \begin{array}{l} u = x\\ dv = \cos \left( {a - x} \right)dx \end{array} \right.$

$\Rightarrow \left\{ \begin{array}{l} du = dx\\ v =  - \sin \left( {a - x} \right) \end{array} \right.$

Do đó

$\begin{array}{l} I = \int\limits_0^{\frac{\pi }{2}} {x\cos \left( {a - x} \right)dx} \\  = \left. { - x\sin \left( {a - x} \right)} \right|_0^{\frac{\pi }{2}} + \int\limits_0^{\frac{\pi }{2}} {\sin \left( {a - x} \right)dx} \\  =  - \frac{\pi }{2}\sin \left( {a - \frac{\pi }{2}} \right) + \left. {\cos \left( {a - x} \right)} \right|_0^{\frac{\pi }{2}}\\  = \frac{\pi }{2}\cos a + \cos \left( {a - \frac{\pi }{2}} \right) - \cos a\\  = \left( {\frac{\pi }{2} - 1} \right)\cos a + \sin a \end{array}$