Integral of \( \displaystyle x \ln{\left(x \right)} \)
Problem 4.726 · easy
Find \( \displaystyle \int x \ln{\left(x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int x \ln{\left(x \right)}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \ln{\left(x \right)} \int x\, dx - \int \frac{\int x\, dx}{x}\, dx \]partsApply integration by parts with u = log(x) and dv = x dx.✓ Proved
- \[ = \frac{x^{2} \ln{\left(x \right)}}{2} - \int \frac{x}{2}\, dx \]substitution algebraSubstitute the derivative of log(x) into the integral. Simplify the integrand.✓ Proved
- \[ = \frac{x^{2} \ln{\left(x \right)}}{2} - \frac{x^{2}}{4} \]antiderivativeEvaluate the final integral.✓ Proved
Answer \( \frac{x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} + C \)
✓ 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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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: fail (style) — Step 3 is labeled "substitution" but it is merely an algebraic simplification of the integrand. The correct label should be "algebra" (or "simplify").qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-09 — Step 3 is labeled "substitution" but it is merely an algebraic simplification of the integrand. The correct label should be "algebra" (or "simplify").qwen3.6:27b-mlx: pass 2026-10-09qwen3.6:27b-mlx: fail (style) 2026-10-09 — Step 3 is labeled 'substitution', but the step performs algebraic simplification of the integrand (substituting the known derivative value and simplifying the product). The label 'substitution' typically refers to the change-of-variable technique (u-sub), which is not applied here. 'Algebra' or 'simplify' would be more accurate labels for evaluating the derivative and simplifying the expression.gpt-oss:20b: fail (style) 2026-10-09 — Step 3 incorrectly labels the operation as "substitution" and the accompanying note is inaccurate; the step merely evaluates the integral of x, which is an algebraic simplification, so the correct label should be "algebra" (or "simplify").
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 2026-10-09 with SymPy 1.14.0.