Evaluate the indefinite integral: integral of x * e^(2x) dx using integration by parts.
Answer and explanation
Correct answer: A. (1/2) x e^(2x) - (1/4) e^(2x) + C
- Step 1: Apply integration by parts formula: integral(u dv) = u*v - integral(v du).
- Step 2: Let u = x => du = dx.
- Step 3: Let dv = e^(2x) dx => v = (1/2) e^(2x).
- Step 4: Substitute: integral = 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) - (1/2) e^(2x) + C' forgets to divide by 2 on the second integration; 'x e^(2x) - e^(2x) + C' ignores the exponential coefficient 2; '(1/4) x^2 e^(2x) + C' integrates both factors simultaneously without parts.
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