Question

How do you integrate hyperbolic trig functions?

Answer 1

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

`sinh(x)=(e^x-e^(-x))/2`
`cosh(x)=(e^x+e^(-x))/2`
`tanh(x)=sinh(x)/cosh(x)=(e^x-e^(-x))/(e^x+e^(-x))`
`coth(x)=cosh(x)/sinh(x)=(e^x+e^(-x))/(e^x-e^(-x))`

From here, it should be reasonably straightforward to show that

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

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

`int sinh(x)dx = int (e^x-e^-x)/2 = int e^x/2-e^(-x)/2dx`
`=e^x/2-(-e^-x)/2 + C` (where `C` is the constant of integration)
`=e^x/2+e^(-x)/2 + C`
`=cosh(x)+C`.

Similarly,
`int cosh(x)dx = int e^x/2+e^(-x)/2dx`
`=e^x/2+(-e^-x)/2 + C`
`=e^x/2-e^-x/2 + C`
`=sinh(x) + C`.