TITLE: Definite integral of the inverse of a function
QUESTION [5 upvotes]: We have $f:\mathbb{R}\to \mathbb{R}$ with $$f\left(x\right)=\frac{\left(-x^3+2x^2-5x+8\right)}{\left(x^2+4\right)}$$
Knowing that the function is bijective, calculate $$\int _{\frac{4}{5}}^2f^{-1}(x)dx$$
How do I solve this? I can't really compute $f^{-1}(x)$. Book tells me to use substitution but I don't really understand what it means.

REPLY [3 votes]: If $y_1 = f(x_1)$ and $y_2 = f(x_2)$, then $$\int_{y_1}^{y_2}f^{-1}(x)\,{\rm d}x = \int_{y_1}^{y_2}f^{-1}(y)\,{\rm d}y = \int_{x_1}^{x_2}xf'(x)\,{\rm d}x,$$because $x = f^{-1}(y) \implies y = f(x) \implies {\rm d}y = f'(x)\,{\rm d}x$. The limits of integration go like this because is $f$ bijective. Now just plug in everything and solve the resulting integral.