If I have a function $h:[0,r] \rightarrow [0,m]$, where $r$, $m>0$.
Assume I have the following integral: $$\int_0^m \frac{dt}{g(t)^\frac{1}{p}},$$ where $g$ is a given function, and $p>1$.
My question is: If I want to make a change of variable $t=h(x)$ to get $$\int_0^r \frac{h'(x)dx}{g(h(x))^\frac{1}{p}},$$ does $h$ need to be only surjective, or the injectivity is essential? And why?