不定积分：e^x(sinx)^2dx

题目

不定积分：e^x(sinx)^2dx

答案

sin²x=(1/2)(1-cos2x)
∫ e^xsin²x dx
=(1/2)∫ e^x(1-cos2x) dx
=(1/2)∫ e^x dx - (1/2)∫ e^xcos2x dx
=(1/2)e^x - (1/2)∫ e^xcos2x dx
下面单独计算
∫ e^xcos2x dx
=∫ cos2x de^x
分部积分
=e^xcos2x + 2∫ e^xsin2xdx
=e^xcos2x + 2∫ sin2xde^x
再分部
=e^xcos2x + 2e^xsin2x - 4∫ e^xcos2x dx
将-4∫ e^xcos2x dx移到左边与左边合并后除以系数
∫ e^xcos2x dx
=(1/5)e^xcos2x + (2/5)e^xsin2x + C
代回到原积分得：
∫ e^xsin²x dx
=(1/2)e^x - (1/2)∫ e^xcos2x dx
=(1/2)e^x - (1/10)e^xcos2x - (1/5)e^xsin2x + C