Let $\int_0^2 f\left(x\right) dx = a+\frac{b}{\log 2}$. Find $a,b$ Let $f$ be a real-valued continuous function on $\mathbb{R}$ such that $2^{f\left(x\right)}+f\left(x\right)=x+1$ for all $x\in \mathbb{R}$. Assume that $\int_0^2 f\left(x\right) dx = a+\dfrac{b}{\log 2}$ with $a,b$ are rational numbers . Find $a,b$.

I have no idea how to use the assumption $2^{f\left(x\right)}+f\left(x\right)=x+1$ for all $x\in \mathbb{R}$ except taking integral of the both sides:
$$\int_0^2 2^{f\left(x\right)}+f\left(x\right) dx=4 $$
Please help me an idea. Thank you.

My solution is here :
I hope this will help you!
(https://i.stack.imgur.com/uCEc3.jpg)
