Definite integrals
Problem 4.125 · medium
Evaluate \( \displaystyle \int_{1}^{4} x + \frac{3}{x} \, dx \).
- By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.Reviewed
- \[ \frac{d}{d x} \left(\frac{x^{2}}{2} + 3 \ln{\left(x \right)}\right) = x + \frac{3}{x} \]F(x) = x**2/2 + 3*log(x) is an antiderivative: its derivative is the integrand.✓ Proved
- \[ - \left. \frac{x^{2}}{2} + 3 \ln{\left(x \right)} \right|_{\substack{ x=1 }} + \left. \frac{x^{2}}{2} + 3 \ln{\left(x \right)} \right|_{\substack{ x=4 }} = \ln{\left(64 \right)} + \frac{15}{2} \]F(b) − F(a).✓ Proved
Answer \( \ln{\left(64 \right)} + \frac{15}{2} \)
✓ Nihil obstat Lines: 2 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | numerical quadrature (mpmath) gives the same value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The arithmetic evaluation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The arithmetic evaluation is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The arithmetic is correct: F(4) - F(1) = (8 + 3ln(4)) - (0.5 + 0) = 7.5 + 3ln(4) = 15/2 + ln(64).gpt-oss:20b: pass 2026-09-28
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_integral, checked 2026-09-28 with SymPy 1.14.0.