∫(x^3+1)(cosx)^2dx
=∫(x^3+1)[(1+cos2x)/2]dx
=(1/2)∫(x^3+1)dx+(1/2)∫cos2xdx+(1/2)∫x^3cos2xdx
=(1/8)x^4+x/2+(1/4)sin2x+(1/4)sin2x *x^3 -(1/4)3x^2sin2xdx
=(1/8)x^4+x/2+(1/4)sin2x+(1/4)sin2x*x^3 +(3/8)cos2x*x^2-(3/4)∫xcos2xdx
=(1/8)x^4+x/2+(1/4)sin2x+(1/4)sin2x*x^3+(3/8)cos2x*x^2-(3/8)sin2x*x-(3/16)cos2x+C