I have to say that I really love that method, I personally use it in a variety of cases, but not always works.

For example, compute The straightforward sense tells us that we must apply integration by parts, but look another way. For and there it is, we have our parameter to construct our double integral with the only reason to reverse integration order and make a little bit simple the problem.

Another example, find Again, follow the same idea, construct a double integral and reverse integration order, you'll get a simple integral to tackle.

How about the famous Dirichlet Integral ? This can also be tackled by introducing the result follows... and so on.

A lot of single integrals can be solved with double integration tricks, but it depends.