The area bounded by the curve f(x) = x + sin x and its inverse function between the ordinates x= 0 to x = 2π, is

Clearly,f:[0,2π]→[0,2π] given byf(x) =x + sin x is a bijection. So, its inverse exists, the graph of f−1(x)  is the mirror image of the graph of f(x) in the line y = x.∴ Required area = 4 (Area of one loop)                              =4∫0π (x+sin⁡x)dx−∫0π xdx=4∫0π sin⁡xdx=−4[cos⁡x]0π=−4[cos⁡π−cos⁡0]=8 sq units