用分部积分法计算∫e^2(sinx)^2dx,我算了好几遍了,和答案不一样啊,

Question

用分部积分法计算∫e^2(sinx)^2 dx,我算了好几遍了,和答案不一样啊,

Answer 1

打错了吧?e^2是e^x吧
∫e^x(sinx)^2dx=∫(sinx)^2de^x=e^x(sinx)^2-∫e^xsin2xdx
∫e^xsin2xdx=∫sin2xde^x=e^xsin2x-2∫e^xcos2xdx=e^xsin2x-2∫cos2xde^x
=e^xsin2x-2e^xcos2x-4∫e^xsin2xdx
所以5∫e^xsin2xdx=e^x(sin2x-2cos2x)
∫e^xsin2xdx=e^x(sin2x-2cos2x)/5
所以∫e^x(sinx)^2dx=e^x(sinx)^2-e^x(sin2x-2cos2x)/5
=e^x(5-cos2x-2sin2x)/10+C