Is it possible to integrate a greatest integer function?

Question

Is there an indefinite integral for this function ?

$$\int [x] dx$$

I know how to integrate it if it was something like this

$$\int_b ^a [x] dx$$

Answer 1

Let $\{t\}=t-[t]$ be the fractional part of $t$. Since $\{t\}$ has period $1$ and its integral over one period is $$ \int_x^{x+1}\{t\}\,\mathrm{d}t=\frac12\tag{1} $$ we see that $$ \int_0^x\{t\}\,\mathrm{d}t=\frac12\{x\}^2+\frac12[x]\tag{2} $$ because $\frac12[x]$ is the integral over the complete periods and $\frac12\{x\}^2$ is the integral over the remaining part of the last period.

Thus, $$ \int\{t\}\,\mathrm{d}t=\frac12\{t\}^2+\frac12[t]+C\tag{3} $$ and $$ \begin{align} \int[t]\,\mathrm{d}t &=\int\left(x-\{x\}\right)\,\mathrm{d}x\\ &=\frac12x^2-\frac12\{x\}^2-\frac12[x]+C\\[3pt] &=[x]x-\frac12[x]^2-\frac12[x]+C\tag{4} \end{align} $$ Here is a plot of $\color{#3F3D9A}{[x]}$ and $\color{#9A3D71}{(4)}$:

Note that for each $k\in\mathbb{Z}$, on $(k,k+1)$, the slope of $(4)$ is $k$. Since $[x]$ is not continuous at the integers, the left and right derivatives of $(4)$ do not match at the integers.