They've used the identity .
Remember that and .
I haven't much experience with hyperbolic functions, having never learned them, but the result of this integral seems to involve one:
Their solution is:
(What's the proper latex command for arsinh?)
I'm not sure how they derived that.
Let . Therefore .Solve .
The integral becomes:
Now use the identity and to get .
Since we require we can rearrange this to get .
We now need to integrate
but using we get that . Combining this with the other identity gives .
We also know that from how we defined u at the very start.
In this case, they've taken .