Concept:Use a suitable substitution to reduce the integrand to the standard form ∫a2−t21dt=sin−1(at)+c.Explanation:Let I=∫9−cos42xsin2xcos2xdx.Multiply the numerator and denominator by 2: I=21∫9−cos42x2sin2xcos2xdx=21∫9−cos42xsin4xdx.Put t=cos22x.Then dt=−4sin2xcos2xdx=−2sin4xdx.Therefore sin4xdx=−21dt.Substituting into I: I=21∫9−t2−21dt=−41∫32−t2dt.Using the standard formula: I=−41sin−1(3t)+c=−41sin−1(3cos22x)+c, where c is the constant of integration.Answer:4−1sin−1(3cos22x)+c