∫(sin5x+cos3xsin2x+sin3xcos2x+cos5x)2sin2xcos2xdx
=∫(sin3x(sin2x+cos2x)+cos3x(cos2x+sin2x))2sin2xcos2xdx
=∫(sin3x+cos3x)2sin2xcos2xdx
=∫(cos3x)2(tan3x+1)2sin2xcos2xdx
=∫(tan3x+1)2tan2xsec2xdx
Put, tan3x+1=t
⇒dt=3tan2xsec2xdx
=∫3(t)2dt=3t−1+C, where C is the constant of integration.
=3(tan3x+1)−1+C