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

* Hướng dẫn giải

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

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