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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. \({\large\int\normalsize} {\text {csch }x \,\text {coth }x\,dx} =\) \( -\text{csch }x + C\)