I have a 6th degree polynomial function with decimal coefficients that I've rounded to make it easier: $$ f(x) = -0.1227 x^6 + 0.0227 x^5 +0.5221 x^4-0.2252 x^3 -0.5113x^2+ 0.3047x +0.1176$$
this polynomial has roots at $x_1= 136.485 $ and $x_2 = 419.239$ and I need to find the inverse of: $$ g(x) = \int_{x_1}^{x} f(x) dx$$ where $x1 \le x \le x2$
Because f(x) is positive over the interval, the integral is always increasing, which means the inverse would be a valid function.
I tried to take the derivative of a composition of functions: $$ \dfrac{d}{dx}\left[g(g^{-1}(x)) = x\right]$$ $$ g'(g^{-1}(x))(g^{-1}(x)' = 1 $$ but I have no idea how I would proceed from there. I've also considered the easier solution in which I just solve for when $g(x)$ is equal to a specific value, but it would be vastly more computationally efficient if I could find a way to solve for an explicit equation for $g^{-1}(x)$.