Consider the question in find a $\int_{x}^{g(x)} f(t)\,dt = 1$. I tried to solve it using ODE.
My attempt: Consider $$H(x) = \int_{x}^{g(x)}f(t)dt$$
for all $x \in [0, 1]$. We look for a $g$ such that $H(x) = 1, \forall x \in [0, 1]$
Thus
\begin{equation} \label{edo} H'(x) = f(g(x))\cdot g'(x) - f(x) = 0 \end{equation}
Finally, we have a separable ODE. By the exercise $g(0) = 1$. Then, we can solve the IVP.
Someone can tell if my attempt is correct? Thank you in advance.