Question Detail

MTH 102 MTH 102 Past Questions Objective Question
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Question

Evaluate the integral ∫ cos3x dx.
Options
A sinx - (sin<sup>3</sup>x)/3 + C
Correct Answer
B cosx + (cos<sup>3</sup>x)/3 + C
C secx - (sec<sup>3</sup>x)/3 + C
D tanx - (tan<sup>3</sup>x)/3 + C
Correct Answer

Option A is the correct answer.

Detailed Explanation

To evaluate ∫ cos3x dx, rewrite cos3x as cos2x * cosx.Then use the identity cos2x = 1 - sin2x:∫ (1 - sin2x) cosx dx.Now, use u-substitution. Let u = sinx, so du = cosx dx.The integral becomes ∫ (1 - u2) du.Integrate with respect to u: u - (u3)/3 + C.Substitute u = sinx back into the expression:sinx - (sin3x)/3 + C.This matches option A.

Hint

Rewrite the odd power of cosine using a Pythagorean identity (cos2x = 1 - sin2x) and then apply u-substitution.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 802 of 930
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