Formulas and Tables

Calculus

Integrals of Hyperbolic Functions

Hyperbolic functions: \(\sinh x,\) \(\cosh x,\) \(\tanh x,\) \(\text{coth }x,\) \(\text{sech }x,\) \(\text{csch }x\)

Argument (independent variable): \(x\)
Real number: \(C\)

  1. Integral involving hyperbolic sine
    \({\large\int\normalsize} {\sinh x\,dx} =\) \( \cosh x + C\)
  2. Integral involving hyperbolic cosine
    \({\large\int\normalsize} {\cosh x\,dx} =\) \( \sinh x + C\)
  3. Integral involving hyperbolic tangent
    \({\large\int\normalsize} {\tanh x\,dx} =\) \( \ln\cosh x + C\)
  4. Integral involving hyperbolic cotangent
    \({\large\int\normalsize} {\text {coth }x\,dx} =\) \(\ln \left| {\sinh x} \right| + C\)
  5. Integral involving hyperbolic secant squared
    \({\large\int\normalsize} {{\text {sech}^2}x\,dx} =\) \( \tanh x + C\)
  6. Integral involving hyperbolic cosecant squared
    \({\large\int\normalsize} {{\text {csch}^2}x\,dx} =\) \( -\text{coth }x + C\)
  7. \({\large\int\normalsize} {\text {sech }x \tanh x\,dx} =\) \( -\text{sech }x + C\)
  8. \({\large\int\normalsize} {\text {csch }x \,\text {coth }x\,dx} =\) \( -\text{csch }x + C\)