∫e^x(sinx)^2dx
问题描述：
∫e^x(sinx)^2dx
答
∫e^x(sinx)^2dx
=1/2∫e^x(1-cos2x)dx
=1/2∫(e^x-e^xcos2x)dx
=1/2∫e^xdx-1/2∫e^xcos2xdx
=1/2e^x-1/2∫e^xcos2xdx
∫e^xcos2xdx
=∫cos2xde^x
=e^xcos2x+2∫e^xsin2xdx
=e^xcos2x+2∫sin2xde^x
=e^xcos2x+2e^xsin2x-4∫e^xcos2xdx
5∫e^xcos2xdx=e^xcos2x+2e^xsin2x=e^x(cos2x+2sin2x)
∫e^xcos2xdx=1/5e^x(cos2x+2sin2x)
所以,原式1/2e^x-1/10e^x(cos2x+2sin2x)+C