\(\int \cos^3 x \:dx\)
\begin{equation*}
\begin{split}
& \int \cos^3 x \:dx\\\\
& \int \cos^2 x \:.\: \cos x \:dx\\\\
& \int \cos^2 x \:.\: \cancel {\cos x} \:\frac{d(\sin x)}{\cancel {\cos x}}\\\\
& {\color {blue} \sin^2 x + \cos^2 x = 1}\\\\
& \int (1 - \sin^2 x) \:d(\sin x)\\\\
& \sin x - \frac{1}{3} \sin^3 x + C
\end{split}
\end{equation*}