Keep numerator inline with rest of equation

Question

In the following example, I would like to to keep the Q''(p) on the same line as the rest of the equation, and put the vinculum and 2 underneath.

\documentclass[12pt,reqno]{amsart}
\usepackage{amsfonts}
\usepackage{amsmath}

\begin{document}
\begin{align}
  \label{eq:3of3}
  \int_{0}^{1} \phi(p)\frac{Q''(p)}{2} \int_{0}^{1} (\Delta p)^{2}\mathcal{P}(\Delta p)\; \mathrm{d}p'\; \mathrm{d}p
\end{align}
\end{document}

The equation currently looks like this:

And I would like something like this:

Answer 1

This "crazy" variant of a fraction is generated by \underline and an operator subscript:

\documentclass[12pt,reqno]{amsart}
\usepackage{amsfonts}
\usepackage{amsmath}

\begin{document}
\begin{align}
  \label{eq:3of3}
  \int_{0}^{1} \phi(p)\mathop{\underline{Q''(p)}}\limits_{\textstyle2}
  \int_{0}^{1} (\Delta p)^{2}\mathcal{P}(\Delta p)\; \mathrm{d}p'\; \mathrm{d}p
\end{align}
\end{document}

For comparison, other variants from the comments:

\documentclass[12pt,reqno]{amsart}
\usepackage{amsfonts}
\usepackage{amsmath}

\begin{document}
\begin{align}
  \label{eq:3of3}
  \int_{0}^{1} \frac{\phi(p)\, Q''(p)}{2}
  \int_{0}^{1} (\Delta p)^{2}\mathcal{P}(\Delta p)\; \mathrm{d}p'\; \mathrm{d}p
\end{align}
\end{document}

\documentclass[12pt,reqno]{amsart}
\usepackage{amsfonts}
\usepackage{amsmath}

\begin{document}
\begin{align}
  \label{eq:3of3}
  \int_{0}^{1} \frac{1}{2}\,\phi(p)\, Q''(p)
  \int_{0}^{1} (\Delta p)^{2}\mathcal{P}(\Delta p)\; \mathrm{d}p'\; \mathrm{d}p
\end{align}
\end{document}

With smaller digits:

\documentclass[12pt,reqno]{amsart}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{textcomp}

\begin{document}
\begin{align}
  \label{eq:3of3}
  \int_{0}^{1} \tfrac{1}{2}\, \phi(p)\, Q''(p)
  \int_{0}^{1} (\Delta p)^{2}\mathcal{P}(\Delta p)\; \mathrm{d}p'\; \mathrm{d}p
\end{align}
\end{document}

Answer 2

With no measurement (available only in display math mode):

\def\crazyfrac#1#2{\raisebox{-\fontdimen8\textfont2}{$\displaystyle\frac{#1}{#2}$}}

It's horrible and, above all, confusing. I decline any responsibility in case of use of this macro in a real document.

Answer 3

Don't do this or @egreg will never forgive me.

\documentclass[12pt,reqno]{amsart}
\usepackage{amsfonts}
\usepackage{amsmath}

\def\horriblefrac#1#2{%
  \setbox0\hbox{$\displaystyle#1$}%
  \setbox2\hbox{$\displaystyle\frac{#1}{#2}$}
  \dimen0\ht0
  \advance\dimen0-\ht2
  \raise \dimen0 \box2}
\begin{document}
\begin{align}
  \label{eq:3of3}
  \int_{0}^{1} \phi(p)\horriblefrac{Q''(p)}{2} \int_{0}^{1} (\Delta p)^{2}\mathcal{P}(\Delta p)\; \mathrm{d}p'\; \mathrm{d}p
\end{align}
\end{document}