Subject: Calculus

List Of Integrals Of Hyperbolic Functions

The following consists of the integrals of the hyperbolic functions and its combinations, products and variations. Note that integrals of hyperbolic functions looks really like the integrals of the ordinary trigonometric function except that there is an additional "h" in the functions and with little extra variations in the signs.

The Page at a Glance

Integrals of the 6 hyperbolic trigonometric functions

  • \int \sinh{u} du = \cosh{u} + C
  • \int \cosh{u} du = \sinh{u} + C
  • \int \tanh{u} du = \ln{\cosh{u}} + C
  • \int \coth{u} du = \ln{|\sinh{u}|} + C
  • \int sechu du = \tan^{-1}{\sinh{u}} + C
  • \int cschu du = \ln{\tanh{\frac{1}{2}u}} + C

Integrals of the variations of the 6 hyperbolic trigonometric functions

  • \int sech^{2}u du = \tanh{u} + C
  • \int csch^{2}u du = -\coth{u} + C
  • \int sechu \tanh{u}du = -sechu + C
  • \int cschu \coth{u}du = -cschu + C
  • \int \sinh^{2}{u} du = -\frac{1}{2}u + \frac{1}{4}\sinh{2u}+C
  • \int \cosh^{2}{u} du = \frac{1}{2}u + \frac{1}{4}\sinh{2u}+C
  • \int \tanh^{2}{u} du = u - \tanh{u} + C
  • \int \coth^{2}{u} du = u - \coth{u} + C

Integrals of hyperbolic trigonometric functions with n - exponents

  • \int \sinh^{n}{u} du = -\frac{1}{n} \sinh^{n-1}{u} \cosh{u} - \frac{n-1}{n} \int \sinh^{n-2}{u}du
  • \int \cosh^{n}{u} du = \frac{1}{n} \cosh^{n-1}{u} \sinh{u} + \frac{n-1}{n} \int \cosh^{n-2}{u}du
  • \int \tanh^{n}{u} du = -\frac{1}{n-1} \tanh^{n-1}{u} + \int \tanh^{n-2}{u}du
  • \int \coth^{n}{u} du = - \frac{1}{n-1} \coth^{n-1}{u} - \int \coth^{n-2}{u}du

Advanced forms of hyperbolic functions (products of sine and cosine)

  • \int \sinh(mu)\sinh(nu) du = \frac{1}{m^2-n^2} \left [m\sinh(nu)\cosh(mu)-n\sinh(mu)\cosh(nu) \right ] + C
  • \int \cosh(mu)\cosh(nu) du = \frac{1}{m^2-n^2} \left [m\sinh(mu)\cosh(nu)-n\sinh(nu)\cosh(mu) \right ] + C
  • \int \cosh(mu)\sinh(nu) du = \frac{1}{m^2-n^2} \left [m\sinh(mu)\cosh(nu)-n\sinh(nu)\cosh(mu) \right ] + C
  • \int u \sinh{u} du = -\sinh{u} + u \cosh{u} + C
  • \int u \cosh{u} du = -\cosh{u} + u \sinh{u} + C