Find the integral where , $f(x)$ is the decimal expansion of $x$.

Question

Let , for each $x\in [0,1)$ $x=0.x_1x_2x_3...$ be the decimal expansion of $x$not eventually all $9's$. Define $f:[0,1)\to \mathbb R$ by $f(x)=x_1$ , the first digit in the expansion. Then $$\int_0^1f(x)\,dx=?$$

I have no idea about this problem. I can not understand that from where I start this problem...Please help..

Answer 1

Hint: take the integral over the intervals $[0,.1)$, $[.1,.2)$, $\dots\;$, $[.9,1)$. What can you say about $f(x)$ on each such interval?