Change Variables in an Integral

Because I can never friggin’ remember. Here’s an example:

K = \int_1^2 x dx = 3/2

OK.

z = e^x
x = ln z
dx = dz / z
K = \int_e^{e^2} (ln z) (dz / z)

Which indeed numerically evaluates to the right thing.

That is, given an integral in terms of x, and wishing to change to f(x):

  1. Let g = (f inverse): such that x = g(z).
  2. Replace integrand using g and dg/dz.
  3. Replace limits a, b with f(a), f(b). Dimensionally, it must be so.