Evaluate the integral using integration by parts: integral of [x * e^(2x)] dx.

Answer and explanation

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

  1. Step 1: Use integration by parts: integral of u dv = uv - integral of v du.
  2. Step 2: Let u = x (so du = dx) and dv = e^(2x) dx (so v = (1/2)e^(2x)).
  3. Step 3: Apply formula: uv - integral of v du = (1/2)x * e^(2x) - integral of (1/2)e^(2x) dx.
  4. Step 4: Evaluate the remaining integral: (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' adds rather than subtracts the second integral term; 'x e^(2x) - e^(2x) + C' ignores the inner factor 2 in the exponential argument; '(1/4)x * e^(2x) - (1/2)e^(2x) + C' misplaces the fractions between the first and second terms.

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