A calculus of the absurd

11.5 Integration of trigonometric functions

11.5.1 Integral of \(\cos ^2(x)\)

Example 11.5.1 Find the value of
\(\seteqnumber{0}{11.}{12}\)
\begin{equation*} \int \cos ^2(x) dx \end{equation*}

Solution: We can’t integrate \(\cos ^2(x)\) directly, so we need to rewrite it first. Of the trigonometric identities, the one which looks most useful is the double-angle formula ⁸²⁸² i.e. \(\cos (2x) = \cos ^2(x) - \sin ^2(x) \implies \cos (2x) = 2\cos ^2(x) - 1 \implies \cos ^2(x) = \frac {\cos (2x) + 1}{2}\)). After rearranging this (see the footnote for details), we can the write that

\begin{align} \int \cos ^2(x) dx &= \int \frac {\cos (2x) + 1}{2} \hspace {2pt} dx \\ &= \frac {1}{2} \int \cos (2x) + 1 \hspace {2pt} dx \\ &= \frac {1}{2} \left [ \frac {1}{2}\sin (2x) + x \right ] + c \\ &= \frac {1}{4} \sin (x) + \frac {1}{2}x \end{align}