Home›Calculus 1›Derivatives and integrals of hyperbolic functions›Problem 2.2319

Derivatives and integrals of hyperbolic functions

Problem 2.2319 · medium

Find \( \displaystyle \int - \sinh{\left(4 x + 1 \right)} \, dx \). (Omit the constant of integration.)

Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
Reviewed
\[ \frac{d}{d x} \left(- \frac{\cosh{\left(4 x + 1 \right)}}{4}\right) = - \sinh{\left(4 x + 1 \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( - \frac{\cosh{\left(4 x + 1 \right)}}{4} + C \)

✓ 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 of sinh as cosh and applies the chain rule factor of 1/4. The verification step confirms the result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the antiderivative of sinh as cosh and applies the chain rule factor of 1/4. The verification step confirms the result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the antiderivative and verifies it by differentiation. The mention of tanh is irrelevant but harmless.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the antiderivative of -sinh(u) as -cosh(u) and properly applies the chain rule for the inner function 4x+1. The verification step confirms the result.
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 2026-10-18 with SymPy 1.14.0.