Calculate ∫_1^e (ln(x))^2 dx.

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Questions & Step-by-Step Solutions

Calculate ∫_1^e (ln(x))^2 dx.

Step 1: Identify the integral we need to calculate: ∫_1^e (ln(x))^2 dx.
Step 2: Use integration by parts, which is a method that helps us integrate products of functions.
Step 3: Choose u = (ln(x))^2 and dv = dx. Then, we need to find du and v.
Step 4: Differentiate u to find du: du = 2(ln(x))(1/x) dx.
Step 5: Integrate dv to find v: v = x.
Step 6: Apply the integration by parts formula: ∫ u dv = uv - ∫ v du.
Step 7: Substitute u, v, du, and dv into the formula: ∫ (ln(x))^2 dx = x(ln(x))^2 - ∫ x * 2(ln(x))(1/x) dx.
Step 8: Simplify the integral: ∫ (ln(x))^2 dx = x(ln(x))^2 - 2∫ ln(x) dx.
Step 9: Now, we need to calculate ∫ ln(x) dx using integration by parts again.
Step 10: For ∫ ln(x) dx, choose u = ln(x) and dv = dx. Then, find du and v.
Step 11: Differentiate u: du = (1/x) dx and integrate dv: v = x.
Step 12: Apply the integration by parts formula again: ∫ ln(x) dx = x ln(x) - ∫ x * (1/x) dx.
Step 13: Simplify: ∫ ln(x) dx = x ln(x) - x + C.
Step 14: Substitute back into the previous equation: ∫ (ln(x))^2 dx = x(ln(x))^2 - 2(x ln(x) - x).
Step 15: Combine the terms: ∫ (ln(x))^2 dx = x(ln(x))^2 - 2x ln(x) + 2x.
Step 16: Now evaluate from 1 to e: Substitute e and 1 into the equation.
Step 17: Calculate the value at e: e(ln(e))^2 - 2e ln(e) + 2e = e(1)^2 - 2e(1) + 2e = e - 2e + 2e = e.
Step 18: Calculate the value at 1: 1(ln(1))^2 - 2(1)ln(1) + 2(1) = 0 - 0 + 2 = 2.
Step 19: Now subtract the two results: e - 2.
Step 20: The final answer is e - 2, which evaluates to 1 when calculated.

Integration by Parts – A technique used to integrate products of functions, based on the formula ∫u dv = uv - ∫v du.
Definite Integrals – Calculating the area under a curve between two specified limits.
Natural Logarithm Properties – Understanding the properties of the natural logarithm function, particularly its behavior and integration.

Calculate ∫_1^e (ln(x))^2 dx.

Practice Questions

Questions & Step-by-Step Solutions

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