Tích phân x \cos d x\) có giá trị bằng

* Hướng dẫn giải

Đặt \(\left\{ \begin{array}{l} u = x\\ dv = \cos \left( {x + \frac{\pi }{4}} \right)dx \end{array} \right. \Rightarrow \left\{ \begin{array}{l} du = dx\\ v = \sin \left( {x + \frac{\pi }{4}} \right) \end{array} \right.\)

Khi đó

\(\begin{aligned} \int_{0}^{\pi} x \cos \left(x+\frac{\pi}{4}\right) d x =\left.\left[x \sin \left(x+\frac{\pi}{4}\right)\right]\right|_{0} ^{\pi}-\int_{0}^{\pi} \sin \left(x+\frac{\pi}{4}\right) d x=\pi \sin \left(\frac{5 \pi}{4}\right)+\left.\left[\cos \left(x+\frac{\pi}{4}\right)\right]\right|^\pi_{0} \\ \end{aligned}\)

\(=-\frac{\pi \sqrt{2}}{2}+\cos \left(\frac{5 \pi}{4}\right)-\cos \left(\frac{\pi}{4}\right)=-\frac{(\pi+2) \sqrt{2}}{2}\)