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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) dx.Now, use a substitution: let u = Sin(x). Then du = Cos(x) dx.The integral becomes:∫ (1 - u2) duIntegrate with respect to u:= u - u3/3 + CSubstitute back u = Sin(x):= Sin(x) - (1/3)Sin3(x) + C.

Hint

Rewrite Cos3(x) as Cos2(x)Cos(x) and use a trigonometric identity before substitution.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 692 of 930
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