This code produces an ankward vertical alignment which involves the integral symbol and the fraction with the square root at the denominator. Why? How can I fix this?
\documentclass[a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\begin{document}
\[
\int_{x_0}^{x(t)} \! \! \! \frac{1}{\sqrt{\frac{2}{m}\bigl
(E_0-V(x)\bigr)}} \, dx = t-t_0
\]
\end{document}
That's how it's supposed to be typeset.
Growing the integral symbol is out of the question, in my opinion. Moving it down is similarly disgraceful.
\documentclass[a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\begin{document}
\noindent
No square root
\[
\int_{x_0}^{x(t)}
\biggl(\frac{2}{m}\bigl(E_0-V(x)\bigr)\biggr)^{\!-1/2}\, dx = t-t_0
\]
Smashed square root
\[
\int_{x_0}^{x(t)}
\frac{1}{\sqrt{\smash[b]{\frac{2}{m}}(E_0-V(x))}} \, dx = t-t_0
\]
Normal
\[
\int_{x_0}^{x(t)}
\frac{1}{\sqrt{\frac{2}{m}\bigl(E_0-V(x)\bigr)}} \, dx = t-t_0
\]
\end{document}
I probably would do
\documentclass[a4paper]{article}
\usepackage{amsmath}
\begin{document}
\[
\int\limits_{x_0}^{x(t)} \frac{1}{\sqrt{\frac{2}{m}\bigl
(E_0-V(\xi)\bigr)}} \, \mathrm{d}\xi = t-t_0
\]
\end{document}
to avoid confusion between the upper limit of the integral, x(t), and the integration variable as well between some quantity d (like a diameter, say) and the differential d. \int\limits makes the integral symbol look higher.
As marmot suggested, I tried using the scalerel package and I like the outcome. It's not perfect, but I think it's acceptable. The integral symbol is finally aligned with the bottom of the fraction.
\documentclass[a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{scalerel}
\begin{document}
\def\x{\frac{1}{\sqrt{\frac{2}{m}\bigl(E_0-V
(\xi)\bigr)}}}
\[
{\scalerel*[2.2ex]{\int}{\x}}_{\!\!\!x_0}^{\,x(t)}\x\,\,
d\xi \, = \, t-t_0
\]
\end{document}
