Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2234

Derivatives and integrals of hyperbolic functions

Problem 2.2234 · medium

Find \( \displaystyle \int x \cosh{\left(x^{2} \right)} \, dx \). (Omit the constant of integration.)

Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
Reviewed
\[ \frac{d}{d x} \frac{\sinh{\left(x^{2} \right)}}{2} = x \cosh{\left(x^{2} \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \frac{\sinh{\left(x^{2} \right)}}{2} + C \)

✓ Nihil obstat Lines: 1 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	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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	a central difference, with sinh and cosh computed from exponentials by mpmath

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the derivative relationship and verifies the result by differentiation. Although the sentence is phrased as a heuristic rather than a formal substitution step, it is mathematically sound and leads to the correct answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the derivative relationship and verifies the result by differentiation. Although the sentence is phrased as a heuristic rather than a formal substitution step, it is mathematically sound and leads to the correct answer.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the derivative relationship and verifies the result by differentiation. Although the sentence is slightly informal, it accurately describes the method used.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the method (substitution) and verifies the result by differentiation. The derivative check confirms the answer is correct.
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/hyperbolic_calculus, checked 2026-10-17 with SymPy 1.14.0.