Explicit inverse of function defined by an integral, in terms of integrand

Question

Consider a strictly increasing smooth(infinitely differentiable) function $f:[a,b]\rightarrow \mathbb{R}$. If we define a function $g(t)$ by $\int_{a}^{t}f(s)ds$, is there in general a nice formula for the inverse of $g$ using the integrand $f$? If there is, is it from multivariable calc or any undergrad math?

Answer 1

Yes - it's a special case of the inverse function theorem. See http://calculus.subwiki.org/wiki/Inverse_function_theorem