Question

求解不定积分∫e^2xsinxdx

Answer 1

设为M,则
M=(1/2)∫sinxd(e^2x)
=(1/2)(e^2x)sinx-(1/2)∫(e^2x)d(sinx)=(1/2)(e^2x)sinx-(1/2)∫(e^2x)cosxdx
=(1/2)(e^2x)sinx-(1/4)∫cosxd(e^2x)
=(1/2)(e^2x)sinx-(1/4)(e^2x)cosx+(1/4)∫(e^2x)d(cosx)
=(1/2)(e^2x)sinx-(1/4)(e^2x)cosx-(1/4)∫(e^2x)sinxdx
=(1/2)(e^2x)sinx-(1/4)(e^2x)cosx-(1/4)M
所以,
(5/4)M=(1/2)(e^2x)sinx-(1/4)(e^2x)cosx
M=(2/5)(e^2x)sinx-(1/5)(e^2x)cosx
=(1/5)(e^2x)(2sinx-cosx)