Question Detail
Question
Evaluate ∫ Cos3 x dx.
Correct Answer
Option A is the correct answer.
Detailed Explanation
To integrate odd powers of cosine, we use the identity cos2x = 1 - sin2x.∫ cos3x dx = ∫ cos2x * cosx dx= ∫ (1 - sin2x) cosx dxNow, use u-substitution. Let u = sinx.Then, du = d/dx(sinx) dx = cosx dx.Substitute u and du into the integral:= ∫ (1 - u2) duIntegrate with respect to u:= u - u3⁄3 + CSubstitute back u = sinx:= sinx - sin3x⁄3 + C.This matches option A.
Hint
Use the trigonometric identity cos2x = 1 - sin2x to rewrite cos3x. Then, use u-substitution with u = sinx.