Problem: Evaluate the integral
Solution: This integral does not fit any box, so we try some substitution. What about if we try to denote the whole denominator by something?
So we can replace the denominator and also dx, but it is not clear what to do with the exponential in the numerator. We get some help from a little algebraic trick:
So we get
Now we could try to replace the root using substitution, but since we now know of the algebraic trick, we could try to start again and better:
This trick with replacing a multiple by a power in the exponential is used quite often, usually it leads (after a substitution) to a rational function.