Slightly bigger integral in exponent

You can use \textstyle.

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\begin{equation*}
e^{i{\textstyle\int} dt\,\frac{1}{2}m\dot x^2}
\end{equation*}

\end{document}

I agree that the integral is small in the superscript, but I think the \textstyle option is too large. For a happy medium I modified the declaration found in exscale.sty to give this

\DeclareFontShape{OMX}{cmex}{m}{n}{%
      <-7>sfixed * cmex7
      <7-8>sfixed * cmex8
      <9>  cmex9
      <10> <10.95> <12> <14.4> <17.28> <20.74> <24.88>cmex10
       }{}

Comparing the two versions of output (including an integral in scriptscriptstyle)

Amsmath sizes:

\DeclareFontShape sizes:

This is specific to computer modern math fonts (because "cmex"), but also affects other parts of math taken from the math extension font. One notable effect is the default rule thickness (fontdimen 8 from the extension font). The effect can be seen on the fractions.

A couple things are going on here. First \big can only be used on delimiters, not on operators which is the source of your error.

Second, the smaller integral sign is a result of loading the amsmath package. Without it, you get the integral sign as shown in your second example. I'm not entirely sure whether the amsmath or the default behavior is the defective one. You'll need to do something like the following to get the correct spacing and sizing of the integral:

e^{ i\mathop {\textstyle\int} \!dt\,\frac{1}{2}m\dot x^2}

I am curious about what's going on below the hood to make \int behave differently and this does merit further investigation.

Edit: Ignore the "solution" above. If you want the default behavior, the way to do this is to specify the cmex10 option when loading amsmath:

\usepackage[cmex10]{amsmath}