Fundamental Theorem of Calculus, Part 1
Problem 4.251 · medium
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{3 x} t^{3} \cos{\left(t \right)} \, dt \).
- By FTC Part 1 and the chain rule, d/dx ∫ from a to u(x) of g(t) dt = g(u(x)) u'(x).
- \[ \frac{d}{d x} 3 x = 3 \]u'(x).✓ Proved
- \[ 81 x^{3} \cos{\left(3 x \right)} \]g(u(x)) u'(x).✓ Proved
Answer \( 81 x^{3} \cos{\left(3 x \right)} \)
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.
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 | the integral computed numerically at x = 0.7 ± 10⁻⁶ and differenced agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution skips the crucial step of evaluating the integrand g(t) = t^3 cos(t) at the upper limit t = 3x. It jumps directly to the final answer without showing that g(3x) = (3x)^3 cos(3x) = 27x^3 cos(3x), which is then multiplied by u'(x) = 3 to get 81x^3 cos(3x). A student would not learn how to apply the formula correctly.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution skips the crucial step of evaluating the integrand g(t) = t^3 cos(t) at the upper limit t = 3x. It jumps directly to the final answer without showing that g(3x) = (3x)^3 cos(3x) = 27x^3 cos(3x), which is then multiplied by u'(x) = 3 to get 81x^3 cos(3x). A student would not learn how to apply the formula correctly.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the Fundamental Theorem of Calculus combined with the chain rule. The derivative of the upper limit is 3, and substituting 3x into the integrand t^3 cos(t) yields 27x^3 cos(3x), which multiplied by 3 gives 81x^3 cos(3x).gpt-oss:20b: pass 2026-10-03
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_part1, checked 2026-10-03 with SymPy 1.14.0.