\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \sqrt{1 - \cos (ax)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
y
 & = & \sqrt{1 - x} \\
& = & - \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} x^{n} \\
& = & 1 - \frac{1}{2} x - \frac{1}{8} x^{2} - \frac{1}{16} x^{3} - \frac{5}{128} x^{4} - \frac{7}{256} x^{5} - \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \sqrt{1 - \cos (ax)} \, \mbox{d}x \\
& = & - \int \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \cos^{n} (ax) \, \mbox{d}x \\
& = & - \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \int \cos^{n} (ax) \, \mbox{d}x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \int \cos^{n} (ax) \, \mbox{d}x \\
& = & - \frac{1}{a} \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \\
 & & \qquad \qquad \qquad \cdot \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: }
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}}
 \left ( ax + \sin (ax) \sum_{i = 0}^{i = \frac{1}{2} n - 1} \frac{2^{2i} (i!)^{2}}{(2i + 1)!} \cos^{2i + 1} (ax) \right ) \\
 n \mbox{ = oneven: }
 \sum_{i = 0}^{i = \frac{1}{2} (n - 1)} \left ( {\frac{1}{2} (n - 1) \atop i} \right ) \frac{(-1)^{i}}{2 i + 1} \sin^{2 i + 1} (ax) \\
 \end{array}} \right .
 + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{1}{a} \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} \\
 & & \qquad \cdot \left \lbrace{
 \begin{array}{l}
 n \mbox{ = even: }
 \frac{n!}{2^{n} \left ( \left ( \frac{1}{2} n \right )! \right )^{2}}
 \left ( ax + \sin (ax) \sum_{i = 0}^{i = \frac{1}{2} n - 1} \frac{2^{2i} (i!)^{2}}{(2i + 1)!} \cos^{2i + 1} (ax) \right ) \\
 n \mbox{ = oneven: }
 \sum_{i = 0}^{i = \frac{1}{2} (n - 1)} \left ( {\frac{1}{2} (n - 1) \atop i} \right ) \frac{(-1)^{i}}{2 i + 1} \sin^{2 i + 1} (ax) \\
 \end{array}} \right .
 + c \\
\end{eqnarray*}
\end{document}
