Rounding to nearest integer symbol in Latex

The mathtools package has a \DeclarePairedDelimiter command which lets you define such macros. I defined a \nint command which encloses its argument between \lfloor and \rceil. The starred version adds a pair of implicit \left...\right, but you may fine-tune the delimiters size using one of \big, \Big, \bigg, \Biggas an optional argument.

\documentclass[10pt,a4paper]{article}
\usepackage{mathtools, nccmath}

\DeclarePairedDelimiter{\nint}\lfloor\rceil

\DeclarePairedDelimiter{\abs}\lvert\rvert

\begin{document}

    \[ \nint{x}\qquad \nint*{\sqrt{x^2 + \mfrac13}}\qquad\nint[\Big]{\sqrt{x^2}} = \abs{x}\]%

\end{document}