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Definite integrals

Problem 4.12 · medium Video

Evaluate \( \displaystyle \int_{1}^{4} x + \frac{5}{x} \, dx \).

By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
\[ \frac{d}{d x} \left(\frac{x^{2}}{2} + 5 \ln{\left(x \right)}\right) = x + \frac{5}{x} \]
F(x) = x**2/2 + 5*log(x) is an antiderivative: its derivative is the integrand.✓ Proved
\[ - \left. \frac{x^{2}}{2} + 5 \ln{\left(x \right)} \right|_{\substack{ x=1 }} + \left. \frac{x^{2}}{2} + 5 \ln{\left(x \right)} \right|_{\substack{ x=4 }} = \ln{\left(1024 \right)} + \frac{15}{2} \]
F(b) − F(a).✓ Proved

Answer \( \ln{\left(1024 \right)} + \frac{15}{2} \)

The graph of f(x) = x + 5/x from x = 1 to x = 4. Rectangles under the curve get thinner until they fill the area, 5log(4) + 15/2.

Lines: 2 proved, 1 not checked. 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	Not checked	—	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: fail (error) — The evaluated integral is 8+10 ln 2, not log(1024)+15/2; the constant term is incorrect.
qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The arithmetic simplification to log(1024) + 15/2 is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The arithmetic simplification to log(1024) + 15/2 is correct.
gpt-oss:20b: fail (error) 2026-09-28 — The evaluated integral is 8+10 ln 2, not log(1024)+15/2; the constant term is incorrect.
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The arithmetic evaluation is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The arithmetic simplification to log(1024) + 15/2 is correct.
gpt-oss:20b: pass 2026-09-26

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.