∫ xcos2x dx

= (1/2)∫ xcos2x d2x

= (1/2)∫ x dsin2x

= (1/2)xsin2x - (1/2)∫ sin2x dx

= (1/2)xsin2x - (1/2)(1/2)(-cos2x) + C

= (1/2)xsin2x + (1/4)cos2x + C