10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-30_515_673_255_639}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve with equation
$$y = ( 1 + 2 \cos 2 x ) ^ { 2 }$$
- Show that
$$( 1 + 2 \cos 2 x ) ^ { 2 } \equiv p + q \cos 2 x + r \cos 4 x$$
where \(p , q\) and \(r\) are constants to be found.
The curve touches the positive \(x\)-axis for the second time when \(x = a\), as shown in Figure 4. The regions bounded by the curve, the \(y\)-axis and the \(x\)-axis up to \(x = a\) are shown shaded in Figure 4.
- Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-32_2255_51_313_1980}