Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2451

Derivatives and integrals of hyperbolic functions

Problem 2.2451 · medium

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

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

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

Lines: 1 proved, 1 not checked. 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	Not checked	—	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: fail (misleading) — The solution claims to use a substitution and derivative identities, but never actually performs the integration; it only verifies the answer by differentiation, which could mislead a student into thinking the integration was carried out correctly.
qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and verifies it by differentiation. The reference to standard hyperbolic derivatives is accurate and sufficient for the context.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the antiderivative and verifies it by differentiation. The reference to standard hyperbolic derivatives is accurate and sufficient for the context.
gpt-oss:20b: fail (misleading) 2026-10-19 — The solution claims to use a substitution and derivative identities, but never actually performs the integration; it only verifies the answer by differentiation, which could mislead a student into thinking the integration was carried out correctly.
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the antiderivative and verifies it by differentiation. The mention of tanh and sech in the first sentence is irrelevant but harmless.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the antiderivative and verifies it by differentiation. The reference to standard hyperbolic derivatives is appropriate.
gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.