Step 1: Identify the integral you need to calculate: ∫_1^e ln(x) dx.
Step 2: Use integration by parts. Let u = ln(x) and dv = dx.
Step 3: Differentiate u to find du: du = (1/x) dx.
Step 4: Integrate dv to find v: v = x.
Step 5: Apply the integration by parts formula: ∫ u dv = uv - ∫ v du.
Step 6: Substitute u, v, du, and dv into the formula: ∫ ln(x) dx = x ln(x) - ∫ x (1/x) dx.
Step 7: Simplify the integral: ∫ ln(x) dx = x ln(x) - ∫ 1 dx.
Step 8: Calculate the integral of 1: ∫ 1 dx = x.
Step 9: Combine the results: ∫ ln(x) dx = x ln(x) - x + C (where C is the constant of integration).
Step 10: Evaluate the definite integral from 1 to e: [x ln(x) - x] from 1 to e.
Step 11: Substitute e into the expression: e ln(e) - e = e - e = 0.
Step 12: Substitute 1 into the expression: 1 ln(1) - 1 = 0 - 1 = -1.
Step 13: Calculate the final result: (0) - (-1) = 1.
Integration of Natural Logarithm – This question tests the ability to integrate the natural logarithm function and apply the Fundamental Theorem of Calculus.
Evaluation of Definite Integrals – It assesses the skill in evaluating definite integrals by substituting the limits of integration correctly.
