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Multiple Choice
Evaluate the indefinite integral. (Remember the constant of integration.)
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Verified step by step guidance
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Step 1: Recognize that the integral involves the natural logarithm function ln(x). To evaluate this, consider using integration by parts, which is based on the formula: ∫u dv = uv - ∫v du.
Step 2: Choose u = ln(x) and dv = dx. This choice is strategic because the derivative of ln(x) simplifies to 1/x, and the integral of dx is x.
Step 3: Compute du and v. From u = ln(x), we find du = (1/x) dx. From dv = dx, we find v = x.
Step 4: Substitute into the integration by parts formula: ∫ln(x) dx = uv - ∫v du = x ln(x) - ∫x * (1/x) dx.
Step 5: Simplify the remaining integral ∫x * (1/x) dx to ∫1 dx, which equals x. Combine the terms to get x ln(x) - x + C, where C is the constant of integration.