How do you integrate hyperbolic trig functions?

1 Answer
Mar 22, 2015

The easiest way to integrate (or differentiate) the hyperbolic functions is to use their definitions:

sinh(x)=exex2
cosh(x)=ex+ex2
tanh(x)=sinh(x)cosh(x)=exexex+ex
coth(x)=cosh(x)sinh(x)=ex+exexex

From here, it should be reasonably straightforward to show that

sinh(x)dx=cosh(x)+C
cosh(x)dx=sinh(x)+C
tanh(x)dx=ln(cosh((x))+C
coth(x)dx=ln(sinh(x))+C

where C is the constant of integration. I will show the first two here:

sinh(x)dx=exex2=ex2ex2dx
=ex2ex2+C (where C is the constant of integration)
=ex2+ex2+C
=cosh(x)+C.

Similarly,
cosh(x)dx=ex2+ex2dx
=ex2+ex2+C
=ex2ex2+C
=sinh(x)+C.