Laplace transforms from the definition
Problem 4.1112 · 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 \( \displaystyle s > -3 \).
- \[ \frac{\partial}{\partial t} \frac{e^{t \left(- s - 3\right)}}{- s - 3} = e^{- 3 t} e^{- s t} \]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
- \[ - \left. \frac{e^{t \left(- s - 3\right)}}{- s - 3} \right|_{\substack{ t=0 }} = \frac{1}{s + 3} \]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 and justifies the boundary term at infinity for the specified domain s > -3.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Fundamental Theorem of Calculus and justifies the boundary term at infinity for the specified domain s > -3.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Fundamental Theorem of Calculus. The condition s > -3 ensures the exponent -(s+3) is negative, guaranteeing the antiderivative vanishes at infinity, 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 applies the definition of the Laplace transform, identifies the antiderivative, and correctly evaluates the limits at infinity and zero for the specified domain s > -3.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.