Question

不定积分：e^x(sinx)^2dx？

Answer 1

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
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