Integral of \( \displaystyle \frac{\ln{\left(x \right)}}{x^{2}} \)
Problem 4.139 · easy
Find \( \displaystyle \int \frac{\ln{\left(x \right)}}{x^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\ln{\left(x \right)}}{x^{2}}\, dx \]integral rewriteStart with the integral of the function. Rewrite the denominator using a negative exponent.✓ Proved
- \[ = - \int \left(- \frac{1}{x^{2}}\right)\, dx - \frac{\ln{\left(x \right)}}{x} \]parts algebraApply integration by parts with u = log(x) and dv = x**(-2) dx. Simplify the integrand in the remaining integral.✓ Proved
- \[ = \int \frac{1}{x^{2}}\, dx - \frac{\ln{\left(x \right)}}{x} \]algebraDistribute the negative sign.✓ Proved
- \[ = - \frac{\ln{\left(x \right)}}{x} - \frac{1}{x} \]antiderivative algebraIntegrate the remaining power of x. Simplify the expression.✓ Proved
Answer \( \frac{- \ln{\left(x \right)} - 1}{x} + C \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies integration by parts and simplifies the resulting expression. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts and simplifies the resulting expression. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts and simplifies the resulting expression. Each step changes only one aspect of the expression and uses valid labels from the allowed vocabulary.gpt-oss:20b: pass 2026-09-29
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-09-29 with SymPy 1.14.0.