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Improper integrals

Problem 4.33 · medium

Evaluate \( \displaystyle \int_{0}^{\infty} e^{- 4 x} \, dx \), or show that it diverges.

The integral is improper at infinity; replace that bound by T and take a limit.
Reviewed
\[ \frac{d}{d x} \left(- \frac{e^{- 4 x}}{4}\right) = e^{- 4 x} \]
An antiderivative, checked by differentiating.✓ Proved
\[ 1 \cdot \frac{1}{4} - \frac{e^{- 4 T}}{4} = \frac{1}{4} - \frac{e^{- 4 T}}{4} \]
The integral with T in place of the bad bound.Reviewed
As T → \infty, this approaches 1/4.
Reviewed
\[ \lim_{T \to \infty}\left(\frac{1}{4} - \frac{e^{- 4 T}}{4}\right) = \frac{1}{4} \]
The limit.✓ Proved

Answer \( \frac{1}{4} \)

✓ Nihil obstat Lines: 2 proved, 3 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
5	✓ 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 over the whole interval agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly sets up the improper integral as a limit, identifies the antiderivative, evaluates the definite integral, and computes the limit accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly sets up the improper integral as a limit, identifies the antiderivative, evaluates the definite integral, and computes the limit accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the improper integral as a limit, identifies the antiderivative, evaluates the definite integral, and computes the limit accurately.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the improper integral, finds the antiderivative, evaluates the definite integral with the variable bound, and correctly computes the limit as the bound approaches infinity.
gpt-oss:20b: pass 2026-09-26

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/improper_integral, checked 2026-10-13 with SymPy 1.14.0.