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MTH 102 MTH 102 Past Questions Objective Question
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Question

Evaluate ∫ x e^(-4x) dx.
Options
A -(1/4)xe^(-4x) - (1/16)e^(-4x) + C
Correct Answer
B (1/4)xe^(-4x) - (1/16)e^(-4x) + C
C -(1/4)xe^(-4x) + (1/16)e^(-4x) + C
D (1/4)xe^(-4x) + (1/16)e^(-4x) + C
Correct Answer

Option A is the correct answer.

Detailed Explanation

This integral requires integration by parts, using the formula ∫ u dv = uv - ∫ v du. Let u = x and dv = e^(-4x) dx. Then, differentiate u to find du: du = dx. Integrate dv to find v: v = ∫ e^(-4x) dx = -(1/4)e^(-4x). Substitute these into the formula: ∫ x e^(-4x) dx = x (-(1/4)e^(-4x)) - ∫ (-(1/4)e^(-4x)) dx. Simplify and integrate the remaining term: = -(1/4)xe^(-4x) + (1/4) ∫ e^(-4x) dx = -(1/4)xe^(-4x) + (1/4) (-(1/4)e^(-4x)) + C = -(1/4)xe^(-4x) - (1/16)e^(-4x) + C.

Hint

Apply integration by parts. Choose u = x and dv = e^(-4x) dx. Remember the LIATE rule for choosing u and dv.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 360 of 930
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