Change Variables in an Integral

2014 April 29

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)`

:

- Let
`g = (f inverse)`

: such that`x = g(z)`

. - Replace integrand using
`g`

and`dg/dz`

. - Replace limits
`a`

,`b`

with`f(a)`

,`f(b)`

. Dimensionally, it must be so.