Lol yes, you might as well just say "in this example, simply solving the integral does the trick".

The whole point of this article is that you can use dimensional analysis to get the form of the answer up to a dimensionless constant. You don't have to know anything about calculus except that dx has the same units as x, and that the integral sign is an additive compounding operation, i.e. it does not change the dimensions of it's argument.

Using a change of variables (the standard trick to solve the integral) still requires you to know 1) how does the differential change under "u-substitution", 2) that the derivative of exp is itself, 3) the chain rule and 4) the fundamental theorem of calculus, which relates definite integrals to the antiderivative of the integrand. In other words, you have to do calculus.

What I meant was that if you write sqrt(a) x = y you get an integral in y (that does not depend on a) times 1/sqrt(a).

It doesn’t require anything except knowing how to change variables in an integral, you don’t have to actually be able to do the integral.