let $f$ be a continuous function defined and integrable in $[a,b]$ in $\mathbb{R}$, and $f^{-1}$ be the compositional inverse of $f$, my question is to deduce the value of :$\displaystyle\int_{a}^{b} f^{-1}(x) dx$ without using the the formula $f^{-1}(x)$ with $\displaystyle\int_{a}^{b} f(x) dx$ is known, then my question here is:
Question: how i get the value of $\displaystyle\int_{a}^{b} f^{-1}(x) dx$ without seeking for $f^{-1}(x)$ formula with $\displaystyle\int_{a}^{b} f(x) dx$ is known ?
I hope you know the formula for $f$ and yes you don't need to actually know the formula for $f^{-1}$. Just put $x = f(t)$ and the integral becomes $$\int_{f^{-1}(a)}^{f^{-1}(b)}tf'(t)\,dt$$ This requires that you know the values of $f^{-1}(a), f^{-1}(b)$. They may be found in certain cases even without deriving a general formula for $f^{-1}$.
But note that this will never be in terms of $\int_{a}^{b}f(x)\,dx$ but rather in terms of $\int_{f^{-1}(a)}^{f^{-1}(b)}f(x)\,dx$.