sin8x−cos8x=(sin2x−cos2x)(sin2x+cos2x)(sin4x+cos4x)
and sin4x+cos4x=(sin2x+cos2x)2−2sin2xcos2x=1−2sin2xcos2x
∴I=∫(1−2sin2xcos2x)sin8x−cos8xdx
=∫(sin4x+cos4x)(sin2x−cos2x)(sin4x+cos4x)dx
=−∫cos2xdx
=−21sin2x+c, where c is the constant of integration.