Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2518

Derivatives and integrals of hyperbolic functions

Problem 2.2518 · medium

Find \( \displaystyle \int \cosh{\left(x - 3 \right)} \, dx \). (Omit the constant of integration.)

Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
Reviewed
\[ \frac{d}{d x} \sinh{\left(x - 3 \right)} = \cosh{\left(x - 3 \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \sinh{\left(x - 3 \right)} + C \)

Lines: 1 proved, 1 reviewed. 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. 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 antiderivative and verifies it by differentiation. Although the problem asks to omit the constant of integration, including it in the stated answer is standard practice and not an error in the derivation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly identifies the antiderivative and verifies it by differentiation. Although the problem asks to omit the constant of integration, including it in the stated answer is standard practice and not an error in the derivation.
gpt-oss:20b: pass 2026-10-28
qwen3.6:27b-mlx: fail (misleading) 2026-10-27 — The solution cites the chain rule derivatives (e.g., (cosh u)' = sinh u) but fails to explicitly apply the chain rule or substitution to handle the inner function (x-3). While the final answer is correct, the reasoning presented is incomplete and could mislead a student into thinking the derivative of cosh(u) is simply sinh(u) without the u' factor, ignoring the necessary integration by substitution steps.
gpt-oss:20b: pass 2026-10-27
qwen3.6:27b-mlx: pass 2026-10-27 — The solution correctly identifies the antiderivative and verifies it by differentiation. The reference to hyperbolic derivative formulas is accurate and sufficient for the method used.
gpt-oss:20b: pass 2026-10-27

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 2026-10-28 with SymPy 1.14.0.