In an earlier section we saw all the derivatives of the hyperbolic functions. Here is a reminder of them.
| Function | Derivative |
|---|---|
| sinh(x) | cosh(x) |
| cosh(x) | sinh(x) |
| tanh(x) | sech2(x) |
| cosech(x) | -coth(x)cosech(x) |
| sech(x) | -tanh(x)sech(x) |
| coth(x) | -cosech2 |
Since integration is the inverse process to differentiation, we can use the results above if we want to integrate any of the functions in the right-hand column. Look at the MathHelp notebook on Integration if you are not completely clear about this.
For example, because the derivative of sinh(x) is cosh(x), that means that the integral of cosh(x) must be sinh(x) PLUS a constant (since the derivative of any constant is zero).
Written mathematically,
Here's a few of that sort for you to try. Write down your answer in each case then click on the integral to see the answer.
So we're OK if we want to integrate any of those particular combinations of hyperbolic functions above. We can extend that to cases where we have a number in front of the x, for example consider the integral of cosh(2x).
We know that the integral of cosh(x) is just sinh(x). In order to differentiate something and get cosh(2x), we might expect that the function we differentiate must have 2x as input as well.
With that in mind we might try differentiating sinh(2x), to see whether we get close to what we're after, namely cosh(2x).
This strategy of trial and error is central to mathematics and is particularly important when you're trying to integrate something.
So, if we differentiate sinh(2x), of course we get 2cosh(2x), because the "2" comes out to the front.
Differentiating sinh(2x) therefore gave us TWICE the answer we were after, so we need to start with HALF of sinh(2x), i.e. 1/2 sinh(2x).
Sure enough, when we differentiate 1/2 sinh(2x), the "2" inside comes out to the front again but this time cancels with the "2" on the bottom of the half, so we end up with just cosh(2x), as we wanted.
This is true whatever number is in front of the x, so for instance the integral of sinh(4x) is 1/4 cosh(4x).
In general then, we can do that for any of those particular combinations of hyperbolic functions that we had above.
Notice that although we can integrate sinh(x) and cosh(x), we don't yet have any way of integrating tanh(x).
In order to integrate tanh(x), we have to use a "trick" involving logs. This is covered in the next section.