Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2515

Derivatives and integrals of hyperbolic functions

Problem 2.2515 · medium

Find \( \displaystyle \int \sinh{\left(2 x + 1 \right)} \, dx \). (Omit the constant of integration.)

Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
Reviewed
\[ \frac{d}{d x} \frac{\cosh{\left(2 x + 1 \right)}}{2} = \sinh{\left(2 x + 1 \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \frac{\cosh{\left(2 x + 1 \right)}}{2} + C \)

✓ 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 antiderivative using the standard derivative rule for hyperbolic cosine and accounts for the chain rule factor. The verification step confirms the result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly identifies the antiderivative using the standard derivative rule for hyperbolic cosine and accounts for the chain rule factor. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-28
qwen3.6:27b-mlx: pass 2026-10-27 — The solution correctly identifies the antiderivative using the standard derivative rule for cosh and the chain rule. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-27
qwen3.6:27b-mlx: pass 2026-10-27 — The solution correctly identifies the antiderivative using the standard derivative rule for hyperbolic cosine and accounts for the chain rule factor of 2. The verification step confirms the result.
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.