Question Detail

MTH 102 MTH 102 Past Questions Objective Question
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Question

Solve ∫ sin x cos x · e^(cos²x) dx.
Options
A -(1/2)e^(cos²x) + C
Correct Answer
B (1/2)e^(cos²x) + C
C -(1/2)e^(-cos²x) + C
D (1/2)e^(-cos²x) + C
Correct Answer

Option A is the correct answer.

Detailed Explanation

To solve ∫ sin x cos x · e^(cos²x) dx, use u-substitution. Let u = cos²x. Then, differentiate u with respect to x: du/dx = 2 cos x (-sin x) = -2 sin x cos x. Therefore, sin x cos x dx = -(1/2) du. The integral transforms into: ∫ e^u (-(1/2) du) = -(1/2) ∫ e^u du = -(1/2)e^u + C. Substituting back u = cos²x, the final answer is -(1/2)e^(cos²x) + C.

Hint

Use u-substitution. Consider what part of the integrand, when differentiated, yields another part of the integrand. Let u = cos²x.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 357 of 930
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