Accumulation functions
Problem 4.1130 · easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} \frac{1}{t} \, dt \) 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
- \[ - \left. \ln{\left(t \right)} \right|_{\substack{ t=1 }} + \left. \ln{\left(t \right)} \right|_{\substack{ t=x }} = \ln{\left(x \right)} \]F(x) = G(x) − G(a).✓ Proved
- \[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]F′(x) = f(x), as FTC Part 1 says.✓ Proved
- \[ \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 0 \]And F(1) = 0.✓ Proved
Answer \( F(x) = \ln{\left(x \right)} \)
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 to evaluate the definite integral, identifying the natural logarithm as the antiderivative and verifying the boundary condition.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the Fundamental Theorem of Calculus to evaluate the definite integral, identifying the natural logarithm as the antiderivative and verifying the boundary condition.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the Fundamental Theorem of Calculus to evaluate the definite integral, identifying the natural logarithm as the antiderivative and verifying the boundary condition.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution uses the Fundamental Theorem of Calculus Part 2 (Evaluation Theorem) to compute the definite integral, but the problem explicitly asks to use the Fundamental Theorem of Calculus Part 1 (which relates the derivative of an integral function to the integrand). Furthermore, the solution assumes the antiderivative is ln(t) without justifying why this specific antiderivative satisfies the initial condition F(1)=0, effectively skipping the logical step that defines the constant of integration.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 2026-10-18 with SymPy 1.14.0.