Question Detail
Question
Evaluate ∫ Cos3 x dx
Correct Answer
Option A is the correct answer.
Detailed Explanation
To integrate Cos3 x, rewrite it using the identity Cos2 x = 1 - Sin2 x:∫ Cos3 x dx = ∫ Cos2 x * Cos x dx = ∫ (1 - Sin2 x) * Cos x dx.Now, use u-substitution. Let u = Sin x. Then, the differential du = Cos x dx.Substitute these into the integral:∫ (1 - u2) du.Integrate with respect to u:= u - (u3/3) + C.Substitute back u = Sin x:= Sin x - (Sin3 x / 3) + C.
Hint
Rewrite the integrand using trigonometric identities. Specifically, express Cos3 x as Cos2 x * Cos x and then use the identity Cos2 x = 1 - Sin2 x. Then, consider u-substitution.