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MTH 102 MTH 102 Past Questions Objective Question
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Question

Evaluate ∫ Cos3 x dx.
Options
A <i>sin x - (1/3)sin<sup>3</sup> x + C</i>
Correct Answer
B <i>cos x + (1/3)cos<sup>3</sup> x + C</i>
C <i>sec x - (1/3)sec<sup>3</sup> x + C</i>
D <i>tan x - (1/3)tan<sup>3</sup> x + C</i>
Correct Answer

Option A is the correct answer.

Detailed Explanation

To integrate Cos3 x, we can rewrite it using the identity Cos2 x = 1 - Sin2 x.∫ Cos3 x dx = ∫ Cos2 x * Cos x dx= ∫ (1 - Sin2 x) Cos x dxNow, use substitution.Let u = Sin x.Then du = d/dx (Sin x) dx = Cos x dx.Substitute u and du into the integral:∫ (1 - u2) duIntegrate term by term:= u - u3/3 + CFinally, substitute u = Sin x back:= Sin x - (Sin3 x)/3 + C.This matches option A.

Hint

For odd powers of sine or cosine, use the Pythagorean identity sin2x + cos2x = 1 to reduce the power, then use substitution.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 757 of 930
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