Accumulation functions
Problem 4.1083 · easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{2}^{x} \frac{1}{t} \, dt \) as a function of \( \displaystyle x \).
- \[ \frac{d}{d t} \ln{\left(t \right)} = \frac{1}{t} \]An antiderivative of the integrand is log(t).✓ Proved
- \[ - \left. \ln{\left(t \right)} \right|_{\substack{ t=2 }} + \left. \ln{\left(t \right)} \right|_{\substack{ t=x }} = \ln{\left(\frac{x}{2} \right)} \]F(x) = G(x) − G(a).✓ Proved
- \[ \frac{d}{d x} \ln{\left(\frac{x}{2} \right)} = \frac{1}{x} \]F′(x) = f(x), as FTC Part 1 says.✓ Proved
- \[ \left. \ln{\left(\frac{x}{2} \right)} \right|_{\substack{ x=2 }} = 0 \]And F(2) = 0.✓ Proved
Answer \( F(x) = \ln{\left(\frac{x}{2} \right)} \)
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 | simplify(lhs - rhs) reduced to 0 |
| 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 from a to three values of x |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral. The steps logically confirm that the proposed function is the correct antiderivative satisfying the initial condition.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral. The steps logically confirm that the proposed function is the correct antiderivative satisfying the initial condition.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral. The steps logically establish that ln(x/2) is the correct antiderivative satisfying the initial condition F(2)=0.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The problem asks to use the Fundamental Theorem of Calculus to write F(x) as a function of x, which implies evaluating the definite integral. The solution incorrectly cites FTC Part 1 (which relates derivatives to integrals) instead of FTC Part 2 (Evaluation Theorem) for the calculation in step 2. Furthermore, step 3 is a verification step, not part of the derivation requested.gpt-oss:20b: pass 2026-10-14
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/ftc_accumulation, checked 2026-10-17 with SymPy 1.14.0.