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Multiple Choice
Evaluate the integral. (Use c for the constant of integration.)
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Verified step by step guidance
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Step 1: Recognize that the integral involves a product of functions, ln(x) and x^2. This suggests using integration by parts, which is based on the formula: ∫u dv = uv - ∫v du.
Step 2: Choose u = ln(x) and dv = x^2 dx. Differentiate u to find du = (1/x) dx, and integrate dv to find v = x^3/3.
Step 3: Substitute into the integration by parts formula: ∫ln(x) x^2 dx = uv - ∫v du. This becomes (ln(x) * x^3/3) - ∫(x^3/3 * (1/x) dx).
Step 4: Simplify the second integral: ∫(x^3/3 * (1/x) dx) = ∫(x^2/3) dx. Integrate x^2/3 to get (x^3/9).
Step 5: Combine the results: The integral becomes (x^3 ln(x)/3) - (x^3/9) + c, where c is the constant of integration.