I want to investigate nested functions; here below are my worked solutions for the integral and derivative of a general nested function.
Let the nested function be defined as
$$y=f(x+f(x+f(x+...))).$$ It can be observed that:
$$f^{-1}y-x=y$$
$$f^{-1}y-y=x$$
$$\frac{\mathrm{dy} }{\mathrm{d} x}(\frac{\mathrm{d} }{\mathrm{d} x}f^{-1}y-1)=1$$
$$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1}{(\frac{\mathrm{d} }{\mathrm{d} x}f^{-1}y-1)}$$
Hence,
$$\int y\,dx$$ $$=\int y(\frac{\mathrm{d} }{\mathrm{d} x}f^{-1}y-1)\, dx$$ $$=yf^{-1}y-\frac{y^2}{2}-\int f^{-1}y\,dx$$
Are my workings correct? Also, are there other interesting results that can be obtained from studying such nested functions?