\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
 & & \int\limits_{x = - \infty}^{x = + \infty} \frac{1}{(a^{2} + x^{2}) (b^{2} + x^{2})} \, \mbox{d}x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\int\limits_{x = - \infty}^{x = + \infty} \frac{1}{(a^{2} + x^{2}) (b^{2} + x^{2})} \, \mbox{d}x
 & = & \left [ \frac{1}{a^{2} - b^{2}} \left ( \frac{1}{b} \arctan \frac{x}{b}
 - \frac{1}{a} \arctan \frac{x}{a} \right ) \right ]_{x = - \infty}^{x = + \infty} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\int\limits_{x = - \infty}^{x = + \infty} \frac{1}{(a^{2} + x^{2}) (b^{2} + x^{2})} \, \mbox{d}x
 & = & \left [ \frac{1}{a^{2} - b^{2}} \left ( \frac{1}{b} \arctan \frac{x}{b}
 - \frac{1}{a} \arctan \frac{x}{a} \right ) \right ]_{x = - \infty}^{x = + \infty} \\
& = & \frac{1}{a^{2} - b^{2}} \left ( \frac{1}{b} \arctan \frac{+ \infty}{b}
 - \frac{1}{a} \arctan \frac{+ \infty}{a} \right ) \\
 & & \qquad \qquad - \, \frac{1}{a^{2} - b^{2}} \left ( \frac{1}{b} \arctan \frac{- \infty}{b}
 - \frac{1}{a} \arctan \frac{- \infty}{a} \right ) \\
& = & \frac{1}{a^{2} - b^{2}} \left ( \frac{1}{b} \frac{\pi}{2} - \frac{1}{a} \frac{\pi}{2} \right )
 - \frac{1}{a^{2} - b^{2}} \left ( \frac{1}{b} \left ( - \frac{\pi}{2} \right ) - \frac{1}{a} \left ( - \frac{\pi}{2} \right ) \right ) \\
& = & \frac{1}{a^{2} - b^{2}} \left ( \frac{\pi}{2b} - \frac{\pi}{2a} + \frac{\pi}{2b} - \frac{\pi}{2a} \right ) \\
& = & \frac{\pi}{a^{2} - b^{2}} \left ( \frac{1}{b} - \frac{1}{a} \right ) \\
& = & \frac{\pi}{a^{2} - b^{2}} \left ( \frac{a}{ab} - \frac{b}{ab} \right ) \\
& = & \frac{\pi}{(a - b) (a + b)} \frac{a - b}{ab} \\
& = & \frac{\pi}{ab (a + b)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\int\limits_{x = - \infty}^{x = + \infty} \frac{1}{(a^{2} + x^{2}) (b^{2} + x^{2})} \, \mbox{d}x
 & = & \left | \, \frac{\pi}{ab (| \, a \, | + | \, b \, |)} \, \right | \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\int\limits_{x = - \infty}^{x = + \infty} \frac{1}{(a^{2} + x^{2}) (b^{2} + x^{2})} \, \mbox{d}x
 & = & \left | \, \frac{\pi}{ab (| \, a \, | + | \, b \, |)} \, \right | \\
\end{eqnarray*}
\end{document}
