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Definite integrals by substitution

Problem 4.311 · medium

Evaluate \( \displaystyle \int_{1}^{e} \frac{4 \ln{\left(x \right)}^{2}}{x}\, dx \).

Let u = log(x); then du = 1/x dx, which is in the integrand up to a constant.
Reviewed
\[ \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 0 \]
The lower limit in u.✓ Proved
\[ \left. \ln{\left(x \right)} \right|_{\substack{ x=e }} = 1 \]
The upper limit in u.✓ Proved
\[ \int\limits_{0}^{1} 4 u^{2}\, du = \frac{4}{3} \]
Integrate in u between the new limits; no back-substitution needed.✓ Proved

Answer \( \frac{4}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature of the original integral

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the substitution method, accurately transforms the limits of integration, and computes the definite integral. The notation 'log' is standard for natural logarithm in this context.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the substitution method, accurately transforms the limits of integration, and computes the definite integral. The notation 'log' is standard for natural logarithm in this context.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the substitution method, accurately transforms the limits of integration, and computes the definite integral correctly.
gpt-oss:20b: pass 2026-10-04

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 generator:structured/definite_substitution, checked 2026-10-04 with SymPy 1.14.0.