Laplace transforms from the definition
Problem 4.1106 · medium
Use the definition \( \displaystyle F(s) = \int_0^\infty e^{-st} f(t)\,dt \) to find the Laplace transform of \( \displaystyle f(t) = e^{3 t} \), for \( \displaystyle s > 3 \).
- \[ \frac{\partial}{\partial t} \frac{e^{t \left(3 - s\right)}}{3 - s} = e^{3 t} e^{- s t} \]An antiderivative of e^(−st) f(t), by differentiating it.✓ Proved
- For s > 3 the antiderivative tends to 0 as t → ∞ (the exponential decay wins).Reviewed
- \[ - \left. \frac{e^{t \left(3 - s\right)}}{3 - s} \right|_{\substack{ t=0 }} = \frac{1}{s - 3} \]So the integral is 0 minus its value at t = 0.✓ Proved
Answer \( F(s) = \frac{1}{s - 3} \)
✓ 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 defining integral computed by mpmath at two values of s |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus for improper integrals, properly justifying the boundary term at infinity and calculating the value at zero.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Fundamental Theorem of Calculus for improper integrals, properly justifying the boundary term at infinity and calculating the value at zero.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Fundamental Theorem of Calculus to the improper integral. The justification for the limit at infinity is sound for s > 3, and the evaluation at t=0 is correct.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the antiderivative, justifies the convergence at infinity for s > 3, and correctly evaluates the definite integral using the Fundamental Theorem of Calculus.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/laplace_transform, checked 2026-10-17 with SymPy 1.14.0.