\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \frac{x}{(a^{2} + x^{2})^{\frac{5}{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & a \tan t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\tan t
 & = & \frac{x}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
t
 & = & \arctan \frac{x}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \frac{x}{(a^{2} + x^{2})^{\frac{5}{2}}} \, \mbox{d}x \\
& = & \int \frac{a \tan t}{(a^{2} + (a \tan t)^{2})^{\frac{5}{2}}} \, \mbox{d}(a \tan t) \\
& = & \int \frac{a \tan t}{(a^{2} + a^{2} \tan^{2} t)^{\frac{5}{2}}} \frac{a}{\cos^{2} t} \, \mbox{d}t \\
& = & \int \frac{\tan t}{a^{3} (1 + \tan^{2} t)^{\frac{5}{2}}} \frac{1}{\cos^{2} t} \, \mbox{d}t \\
& = & \int \frac{1}{a^{3} \left ( \frac{\cos^{2} t}{\cos^{2} t} + \frac{\sin^{2} t}{\cos^{2} t} \right )^{\frac{5}{2}}} \frac{\sin t}{\cos^{3} t}
 \, \mbox{d}t \\
& = & \int \frac{1}{a^{3} \left ( \frac{\cos^{2} t + \sin^{2} t}{\cos^{2} t} \right )^{\frac{5}{2}}} \frac{\sin t}{\cos^{3} t} \, \mbox{d}t \\
& = & \int \frac{1}{a^{3} \left ( \frac{1}{\cos^{2} t} \right )^{\frac{5}{2}}} \frac{\sin t}{\cos^{3} t} \, \mbox{d}t \\
& = & \int \frac{1}{a^{3} \left ( \frac{1}{\cos^{5} t} \right )} \frac{\sin t}{\cos^{3} t} \, \mbox{d}t \\
& = & \int \frac{1}{a^{3}} \sin t \cos^{2} t \, \mbox{d}t \\
& = & - \frac{1}{3 a^{3}} \cos^{3} t + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\cos \alpha
 & = & \frac{1}{\sqrt{1 + \tan^{2} \alpha}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{1}{3 a^{3}} \cos^{3} t + c \\
& = & - \frac{1}{3 a^{3}} \left ( \frac{1}{\sqrt{1 + \tan^{2} t}} \right )^{3} + c \\
& = & - \frac{1}{3 a^{3}} \frac{1}{(1 + \tan^{2} t)^{\frac{3}{2}}} + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{1}{3 a^{3}} \frac{1}{(1 + \tan^{2} t)^{\frac{3}{2}}} + c \\
& = & - \frac{1}{3 a^{3}} \frac{1}{(1 + \tan^{2} \arctan \frac{x}{a})^{\frac{3}{2}}} + c \\
& = & - \frac{1}{3 a^{3}} \frac{1}{\left ( 1 + \left ( \frac{x}{a} \right )^{2} \right )^{\frac{3}{2}}} + c \\
& = & - \frac{1}{3} \frac{1}{(a^{2} + x^{2})^{\frac{3}{2}}} + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{\mbox{d}(F (x))}{\mbox{d}x} \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( - \frac{1}{3} \frac{1}{(a^{2} + x^{2})^{\frac{3}{2}}} + c \right ) \\
& = & - \frac{1}{3} \left ( - \frac{3}{2} \right ) \frac{1}{(a^{2} + x^{2})^{\frac{5}{2}}} (2x) + 0 \\
& = & \frac{x}{(a^{2} + x^{2})^{\frac{5}{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{1}{3} \frac{1}{(a^{2} + x^{2})^{\frac{3}{2}}} + c \\
\end{eqnarray*}
\end{document}
