1． $\displaystyle{\int_{0}^{\frac{\pi}{2}} x \cos 2 x \mathrm{~d} x}$ ．
选择分部积分公式
令 u = x, dv = cos(2x) dx，则 du = dx，v = (1/2) sin(2x)。
应用分部积分公式
∫ x cos(2x) dx = (1/2)x sin(2x) - ∫ (1/2) sin(2x) dx = (1/2)x sin(2x) + (1/4) cos(2x) + C
代入上下限计算定积分
∫_{0}^{π/2} x cos(2x) dx = [(1/2)x sin(2x) + (1/4) cos(2x)]_{0}^{π/2} = (1/2)(π/2) sin(π) + (1/4) cos(π) - [0 + (1/4) cos(0)] = 0 + (1/4)(-1) - (1/4)(1) = -1/4 - 1/4 = -1/2