∫sin2x/(cosx+sinx2)dx

Question

如题所述

Answer 1

∫sin2x/(cosx+(sinx)^2)dx
=2∫sinxcosx/(-(cosx)^2+cosx+1)dx
=2∫cosx/(-(cosx)^2+cosx+1) ( sinxdx)

consider
-(cosx)^2+cosx+1 = 5/4-(cosx - 1/2)^2
let
cosx -1/2 = (√5/2)siny
-sinx dx = (√5/2)cosy dy
sinxdx =-(√5/2)cosy dy

∫sin2x/(cosx+(sinx)^2)dx
=2∫cosx/(-(cosx)^2+cosx+1) ( sinxdx)
=2∫{ ((√5/2)siny+1/2) /[(5/4)(cosy)^2] } (-(√5/2)cosy dy)
=-(8√5/25)∫ (√5siny+1)/cosy dy
=-(8√5/25)∫ (√5tany+secy ) dy
=-(8√5/25)[ -5ln|cosy|+ln|secy+tany| ] + C