[Math] Find the area between $x + \sin x$ and its inverse

Question

I am confused this problem is really hard for me . I don't know how to calculate this function's inverse but my teacher said that this can be done without finding out the inverse of the function . Please help .

Answer 1

Hint The function and its inverse will be symmetric about the line $y=x$. so you can mange your calculation to computing a simple integral.

For example if the bounds are $x=0$ and $x=\pi$ then the wanted area is just double the area between $y= x+\sin x$ and $y=x$