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Derivatives and integrals of hyperbolic functions practice problems

Derivatives and antiderivatives of sinh, cosh, tanh and the inverse hyperbolic functions. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 2.2234

medium
Find \( \displaystyle \int x \cosh{\left(x^{2} \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \frac{\sinh{\left(x^{2} \right)}}{2} = x \cosh{\left(x^{2} \right)} \]
    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. The full receipt · How we verify

Problem 2.2236

medium
Find \( \displaystyle \int - \sinh{\left(4 x - 1 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \left(- \frac{\cosh{\left(4 x - 1 \right)}}{4}\right) = - \sinh{\left(4 x - 1 \right)} \]
    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. The full receipt · How we verify

Problem 2.2238

medium
Find \( \displaystyle \int \cosh{\left(3 x + 3 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \frac{\sinh{\left(3 x + 3 \right)}}{3} = \cosh{\left(3 x + 3 \right)} \]
    Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\sinh{\left(3 x + 3 \right)}}{3} + C \)

✓ 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. The full receipt · How we verify

Problem 2.2241

medium
Differentiate \( \displaystyle f(x) = \sinh{\left(x^{2} \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \sinh{\left(x^{2} \right)} = 2 x \cosh{\left(x^{2} \right)} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = 2 x \cosh{\left(x^{2} \right)} \)

✓ 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. The full receipt · How we verify

Problem 2.2242

medium
Differentiate \( \displaystyle f(x) = x \cosh{\left(x \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} x \cosh{\left(x \right)} = x \sinh{\left(x \right)} + \cosh{\left(x \right)} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = x \sinh{\left(x \right)} + \cosh{\left(x \right)} \)

✓ 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. The full receipt · How we verify

Problem 2.2243

medium
Differentiate \( \displaystyle f(x) = \tanh{\left(e^{x} \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \tanh{\left(e^{x} \right)} = \frac{e^{x}}{\cosh^{2}{\left(e^{x} \right)}} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = \frac{e^{x}}{\cosh^{2}{\left(e^{x} \right)}} \)

✓ 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. The full receipt · How we verify

Problem 2.2312

medium
Differentiate \( \displaystyle f(x) = \sinh{\left(e^{x} \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \sinh{\left(e^{x} \right)} = e^{x} \cosh{\left(e^{x} \right)} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = e^{x} \cosh{\left(e^{x} \right)} \)

✓ 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. The full receipt · How we verify

Problem 2.2313

medium
Find \( \displaystyle \int \cosh{\left(2 x - 3 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \frac{\sinh{\left(2 x - 3 \right)}}{2} = \cosh{\left(2 x - 3 \right)} \]
    Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\sinh{\left(2 x - 3 \right)}}{2} + C \)

✓ 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. The full receipt · How we verify

Problem 2.2315

medium
Differentiate \( \displaystyle f(x) = \operatorname{atanh}{\left(\frac{x}{2} \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \operatorname{atanh}{\left(\frac{x}{2} \right)} = - \frac{2}{x^{2} - 4} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = - \frac{2}{x^{2} - 4} \)

✓ 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. The full receipt · How we verify

Problem 2.2318

medium
Differentiate \( \displaystyle f(x) = \operatorname{asinh}{\left(3 x \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \operatorname{asinh}{\left(3 x \right)} = \frac{3}{\sqrt{9 x^{2} + 1}} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = \frac{3}{\sqrt{9 x^{2} + 1}} \)

✓ 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. The full receipt · How we verify

Problem 2.2319

medium
Find \( \displaystyle \int - \sinh{\left(4 x + 1 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \left(- \frac{\cosh{\left(4 x + 1 \right)}}{4}\right) = - \sinh{\left(4 x + 1 \right)} \]
    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. The full receipt · How we verify

Problem 2.2320

medium
Find \( \displaystyle \int \cosh{\left(x + 2 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \sinh{\left(x + 2 \right)} = \cosh{\left(x + 2 \right)} \]
    Differentiating the answer returns the integrand.✓ Proved
Answer \( \sinh{\left(x + 2 \right)} + C \)

✓ 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. The full receipt · How we verify

Problem 2.2348

medium
Differentiate \( \displaystyle f(x) = \operatorname{asinh}{\left(x \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \operatorname{asinh}{\left(x \right)} = \frac{1}{\sqrt{x^{2} + 1}} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = \frac{1}{\sqrt{x^{2} + 1}} \)

✓ 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. The full receipt · How we verify

Problem 2.2349

medium
Find \( \displaystyle \int \cosh{\left(x - 2 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \sinh{\left(x - 2 \right)} = \cosh{\left(x - 2 \right)} \]
    Differentiating the answer returns the integrand.✓ Proved
Answer \( \sinh{\left(x - 2 \right)} + C \)

✓ 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. The full receipt · How we verify

Problem 2.2353

medium
Find \( \displaystyle \int \sinh{\left(4 x \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \frac{\cosh{\left(4 x \right)}}{4} = \sinh{\left(4 x \right)} \]
    Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\cosh{\left(4 x \right)}}{4} + C \)

✓ 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. The full receipt · How we verify

Problem 2.2355

medium
Find \( \displaystyle \int \sinh{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \frac{\cosh{\left(3 x - 1 \right)}}{3} = \sinh{\left(3 x - 1 \right)} \]
    Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\cosh{\left(3 x - 1 \right)}}{3} + C \)

✓ 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. The full receipt · How we verify

Problem 2.2358

medium
Differentiate \( \displaystyle f(x) = \sinh{\left(\sqrt{x} \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \sinh{\left(\sqrt{x} \right)} = \frac{\cosh{\left(\sqrt{x} \right)}}{2 \sqrt{x}} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = \frac{\cosh{\left(\sqrt{x} \right)}}{2 \sqrt{x}} \)

✓ 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. The full receipt · How we verify

Problem 2.2445

medium
Differentiate \( \displaystyle f(x) = \tanh{\left(\sqrt{x} \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \tanh{\left(\sqrt{x} \right)} = \frac{1}{2 \sqrt{x} \cosh^{2}{\left(\sqrt{x} \right)}} \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = \frac{1}{2 \sqrt{x} \cosh^{2}{\left(\sqrt{x} \right)}} \)

✓ 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. The full receipt · How we verify

Problem 2.2515

medium
Find \( \displaystyle \int \sinh{\left(2 x + 1 \right)} \, dx \). (Omit the constant of integration.)
  1. Use (cosh u)′ = sinh u, (sinh u)′ = cosh u, (tanh u)′ = sech²u, with a substitution for the inside.
    Reviewed
  2. \[ \frac{d}{d x} \frac{\cosh{\left(2 x + 1 \right)}}{2} = \sinh{\left(2 x + 1 \right)} \]
    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. The full receipt · How we verify

Problem 2.2232

easy
Differentiate \( \displaystyle f(x) = - \tanh{\left(x + 2 \right)} \).
  1. (sinh u)′ = cosh u·u′, (cosh u)′ = sinh u·u′, (tanh u)′ = sech²u·u′, (sinh⁻¹u)′ = u′/√(1 + u²), (tanh⁻¹u)′ = u′/(1 − u²).
    Reviewed
  2. \[ \frac{d}{d x} \left(- \tanh{\left(x + 2 \right)}\right) = \tanh^{2}{\left(x + 2 \right)} - 1 \]
    Apply the rule (and the chain rule).✓ Proved
Answer \( f'(x) = \tanh^{2}{\left(x + 2 \right)} - 1 \)

✓ 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. The full receipt · How we verify

More derivatives and integrals of hyperbolic functions practice

Differentiate \( \displaystyle f(x) = \sinh{\left(x \right)} \cosh{\left(x \right)} \).
Problem 2.2237easy✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{atanh}{\left(\frac{x}{4} \right)} \).
Problem 2.2239easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cosh{\left(4 x - 1 \right)} \).
Problem 2.2244easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cosh{\left(3 x - 3 \right)} \).
Problem 2.2245easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \tanh{\left(3 x + 3 \right)} \).
Problem 2.2246easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \operatorname{asinh}{\left(x \right)} \).
Problem 2.2311easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \operatorname{asinh}{\left(4 x \right)} \).
Problem 2.2317easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cosh{\left(x - 1 \right)} \).
Problem 2.2346easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \tanh{\left(4 x + 2 \right)} \).
Problem 2.2350easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \operatorname{asinh}{\left(4 x \right)} \).
Problem 2.2351easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cosh{\left(3 x + 3 \right)} \).
Problem 2.2352easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \operatorname{asinh}{\left(2 x \right)} \).
Problem 2.2354easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cosh{\left(2 x \right)} \).
Problem 2.2356easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cosh{\left(x - 3 \right)} \).
Problem 2.2360easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \sinh{\left(3 x + 1 \right)} \).
Problem 2.2447easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \operatorname{asinh}{\left(2 x \right)} \).
Problem 2.2449easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sinh{\left(3 x + 1 \right)} \).
Problem 2.2450easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \cosh{\left(3 x - 1 \right)} \).
Problem 2.2452easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sinh{\left(2 x + 3 \right)} \).
Problem 2.2512easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \sinh{\left(x + 2 \right)} \).
Problem 2.2513easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sinh{\left(x - 3 \right)} \).
Problem 2.2514easy✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \tanh{\left(3 x - 1 \right)} \).
Problem 2.2517easy✓ Every equation proved
Find \( \displaystyle \int \tanh{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2233medium✓ Every equation proved
Find \( \displaystyle \int \tanh{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2235medium✓ Every equation proved
Find \( \displaystyle \int - \tanh{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2240medium✓ Every equation proved
Find \( \displaystyle \int - \tanh{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2314medium✓ Every equation proved
Find \( \displaystyle \int \cosh{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2316medium✓ Nihil obstat
Find \( \displaystyle \int - \tanh{\left(x \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2347medium✓ Every equation proved
Find \( \displaystyle \int \cosh{\left(2 x + 2 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2357medium✓ Nihil obstat
Find \( \displaystyle \int \cosh{\left(x + 3 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2359medium✓ Nihil obstat
Find \( \displaystyle \int \cosh{\left(2 x - 2 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2444medium✓ Nihil obstat
Find \( \displaystyle \int \sinh{\left(4 x - 2 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2446medium✓ Nihil obstat
Find \( \displaystyle \int \tanh{\left(x \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2448medium✓ Every equation proved
Find \( \displaystyle \int \sinh{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2451medium✓ Every equation proved
Find \( \displaystyle \int - \sinh{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2453medium✓ Nihil obstat
Find \( \displaystyle \int \cosh{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2510medium✓ Nihil obstat
Find \( \displaystyle \int - \sinh{\left(2 x + 3 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2511medium✓ Nihil obstat
Find \( \displaystyle \int \cosh{\left(2 x + 1 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2516medium✓ Nihil obstat
Find \( \displaystyle \int \cosh{\left(x - 3 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2518medium✓ Every equation proved
Find \( \displaystyle \int - \sinh{\left(3 x - 2 \right)} \, dx \). (Omit the constant of integration.)
Problem 2.2519medium✓ Nihil obstat