What kind of integral is this?

With the help of the bigints package (cf. Big integral sign), and some negative space, using Steven's code as a starting point.

\documentclass{article} 
\usepackage{bigints}
\begin{document}
\[
\mathit{Life} = \hspace{-0.8em} \bigint_{\mathit{birth}}^{\mathit{death}}\hspace{-2.5em}\frac{\mathit{happiness}}{\mathit{time}} \Delta \mathit{time}
\]
\end{document}

Using bigints is a good start. Negative kerning is then needed around the equality and before the fraction.

\documentclass{article}

\usepackage[osf]{Baskervaldx} % tosf in text, tlf in math
\usepackage[baskervaldx,vvarbb]{newtxmath}
\usepackage{bigints}

\begin{document}

\begin{displaymath}
  \textit{Life} \mkern-2mu = \mkern-18mu
  \bigints_{\textit{birth}}^{\textit{death}} \mkern-42mu
  \frac{\textit{happiness}}{\textit{time}} \,\Delta \textit{time}
\end{displaymath}

\end{document}

Here's a start.

\documentclass{article} 
\begin{document}
\[
\mathit{Life} = \int_\mathit{birth}^\mathit{death}\frac{\mathit{happiness}}{\mathit{time}} \Delta \mathit{time}
\]
\end{document}

More refined:

\documentclass{article} 
\usepackage{scalerel,mathtools}
\begin{document}
\[
\mathit{Life} {\,=} \!\!\!\!\vcenter{\hbox{%
  $\displaystyle{\scaleto[2em]{\int}{60pt}}_{%
    \mkern-27mu\mathit{birth}}^{\mathit{death}}$}}\mkern-44mu%
      \frac{\mathit{happiness}}{\mathit{time}}\,\Delta\,\mathit{time}
\]
\end{document}