We can prove that if $g$ is strictly increasing and absolutely continuous on $[a,b]$, and $f$ is a nonnegative integrable function on $[c,d]$, where $c=g(a), d=g(b)$, then
$ \int_c^d f(y)dy = \int_a^b f(g(x))g'(x)dx $
I wonder if the formula remains valid if $g$ is only increasing, not necessarily strict?
I think it is true, but I'm not quite convinced.