I have been working out some cases where I can't integrate through trigonometric substitutions (or at least, not easily) but I can using hyperbolic functions. See if you can solve
$ \int x^2\sqrt{x^2+1}dx $
Special points if you can solve it using trig functions.
The method and thought process
Why couldn't you substitute x^2+1 for u and say x^2 = u-1. then, distribute and just use the power rule. There is no need for trig substitution for this. - G Briz
That works wonder if the first part of the integral is x to the third power, but in this case, you end up with an uneliminatable x in the derivative of u. -- John Mason
