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\documentclass{article}
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The following sum is a {\em display equation}:

\begin{equation}
%
\int_{-\infty}^\infty u \cos{u} \mbox{\em du} = \cos u + u \sin u
%
\end{equation}

The following is an in-line formula:
$\int_{-\infty}^\infty u \cos{u} \mbox{\em du} = \cos u + u \sin u$.

\[
x = \frac {a+b+c/2} {\alpha_i}
\]

\[
x = \frac {x^{2y}} {\log{x+y}}
\]

\[
f(x) = \left\{
  \begin{array}{ll}
    a & \mbox{if $x<0$}
\\
    c & \mbox{otherwise}
  \end{array} \right.
\]

\end{document}