\begin{equation*}
\begin{split}
& \int 2 \cos^2 x \:dx\\\\
& {\color {blue} \cos^2 x = \frac{1+\cos 2x}{2} }\\\\
& \int \cancel {2} \:.\: \frac{1+\cos 2x}{\cancel {2}} \:dx\\\\
& \int 1+\cos 2x\:dx\\\\
& \int 1 \: dx + \int \cos 2x \: \frac{d(2x)}{2}\\\\
& x+\frac{1}{2}\sin 2x +C
\end{split}
\end{equation*}
