Integral of \( \displaystyle \frac{\ln{\left(x \right)}}{x} \)
Problem 4.183 · easy Video
Find \( \displaystyle \int \frac{\ln{\left(x \right)}}{x} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\ln{\left(x \right)}}{x}\, dx \]integral algebraStart with the integral of the given function. Rewrite the fraction as a product.✓ Proved
- \[ = \frac{\ln{\left(x \right)}^{2}}{2} \]antiderivativeThe antiderivative of log(x)/x is (log(x)^2)/2.✓ Proved
Answer \( \frac{\ln{\left(x \right)}^{2}}{2} + C \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
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 |
| 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 |
| 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: fail (misleading) — Step 3 jumps directly to the final answer without showing the substitution $u = \log(x)$, which is the standard method for this integral. Labeling this jump as 'antiderivative' is misleading because it implies the result is a basic known form, whereas it requires a non-trivial substitution step that is omitted.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-03 — Step 3 jumps directly to the final answer without showing the substitution $u = \log(x)$, which is the standard method for this integral. Labeling this jump as 'antiderivative' is misleading because it implies the result is a basic known form, whereas it requires a non-trivial substitution step that is omitted.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the substitution $u = \log(x)$ implicitly in step 3, and the label 'antiderivative' is appropriate for recognizing the standard form. Step 2 is a valid algebraic rewrite.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (misleading) 2026-09-29 — Step 3 jumps directly to the final answer without showing the substitution $u = \log(x)$, which is the standard method for this integral. Labeling this jump as 'antiderivative' is misleading because it implies the result is a known basic form, whereas it actually requires a substitution step that is missing from the solution.gpt-oss:20b: fail (style) 2026-09-29 — Step 3 applies a substitution (u=log x, du=1/x dx) but is labeled "antiderivative". The correct rule name from the allowed vocabulary is "substitution".
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 2026-10-03 with SymPy 1.14.0.