(cosx)^2=(1+cos2x)/2
(cosx)^4=((1+cos2x)/2)^2=(1+2cos2x+cos^2 2x)/4
=(1+2cos2x+(1+cos4x)/2)/4
x*(cosx)^4=3x/8 +xcos2x /2+ xcos4x /8
∫3x/8 +xcos2x /2+ xcos4x /8 dx
=3x^2/16 +∫xcos2x /2 dx+∫xcos4x /8 dx
=3x^2/16+∫x/4 dsin2x +∫x/32 dsin4x
=3x^2/16+xsin2x/4 -∫sin2x /4 dx+xsin4x /32 -∫sin4x /32 dx
=3x^2/16+xsin2x /4+cos2x /8 +xsin4x/32 +cos4x /128+ c