# Too innocent of an integral

Calculus Level 3

Let $$f(x) = x + e^x - 1$$, and let $$f^{-1} (x)$$ denote the inverse function of $$f(x)$$.

Then the integral $$\displaystyle \int_{e}^{1 + e^2} f^{-1} (x) \, dx$$ evaluates to $k_0 + k_1 e^1 + k_2 e^2 + \cdots + k_n e^n ,$ where $$k_0, k_1, \ldots, k_n$$ are rational numbers and $$e\approx 2.718$$ is Euler's number.

If the sum $$k_0 + k_1 + k_2 + \cdots + k_n$$ can be expressed as $$\frac{A}{B},$$ where $$A$$ and $$B$$ are coprime positive integers, what is $$A+B+n?$$

×