Let's look at some examples:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\diff}{\mathop{}\!d}

\begin{document}
\begin{gather*}
\int_{-\infty}^{+\infty}               e^{-i2\pi xt}f(x)\diff x   \\
\int_{-\infty}^{+\infty}  \!           e^{-i2\pi xt}f(x)\diff x   \\
\int_{-\infty}^{+\infty}  \! \!        e^{-i2\pi xt}f(x)\diff x   \\
\int_{-\infty}^{+\infty}  \! \! \!     e^{-i2\pi xt}f(x)\diff x   \\
\int_{-\infty}^{+\infty}  \! \! \! \!  e^{-i2\pi xt}f(x)\diff x
\end{gather*}
\end{document}




The first and the second lines seem right; perhaps, due to the low ‘e’ a negative shift is good, so the second line could be preferred.

In no case should the integrand go below the integration bounds. So, starting from the third line the result are on the worse side.

Making integrals look prettier with negative spacing for publication
As a non-native speaker, I do not fully understand the meaning of the term "to become controversial" in the following text, which I have to translate into German:


  Where translators have made their presence felt they have become controversial, with those who characteristically intervene, seeking to domesticate texts, contrasted with those who foreignise, supposedly allowing other languages to alter our own.

