TITLE: Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse
QUESTION [5 upvotes]: What are some interesting functions I can use to demonstrate this integration trick:
$$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$
I would like to know of some interesting functions where this trick is not obvious. EDIT: The functions I am receiving are the obvious ones like $f(x)=x$, which I don't want. :)

REPLY [3 votes]: Take $$f(x)=x+\sin x$$
Now, if you are asked to find $$\int_{0}^{\pi}f^{-1}(x)dx^{**}$$Do you know $f^{-1*}$,No!
Puzzled! Use the identity:
$$\int_{0}^{\pi}f^{-1}(x)dx=\pi^2-\int_{0}^{\pi}f(x)dx=\frac{\pi^2}2-2$$ 
Otherwise put you can use it anywhere you can't get $f^{-1}$ in explicit forms.
*in explicit form
**$\small f(0)=0\wedge f(\pi)=\pi$