Using the method of integration by parts, evaluate the integral: integral of x * e^(2x) dx.

Answer and explanation

Correct answer: A. (1/2) x e^(2x) - (1/4) * e^(2x) + C

  1. Step 1: Integration by parts formula: integral u dv = u * v - integral v du.
  2. Step 2: Let u = x => du = dx. Let dv = e^(2x) dx => v = (1/2) * e^(2x).
  3. Step 3: Substitute: integral x e^(2x) dx = x [(1/2) e^(2x)] - integral [(1/2) e^(2x)] dx = (1/2) x e^(2x) - (1/4) * e^(2x) + C.
Answer
(1/2) x e^(2x) - (1/4) * e^(2x) + C.
Why not the others: '(1/2) x e^(2x) + (1/4) e^(2x) + C' has an incorrect plus sign in the integration by parts formula; 'x e^(2x) - (1/2) e^(2x) + C' fails to divide by 2 when integrating e^(2x); '(1/4) x e^(2x) - (1/2) e^(2x) + C' inverts the coefficient weights.

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