Question

What is the integral of $\cos \left(2\theta \right)$?

Answer 1

Integral of $\cos \left(2\theta \right)$.

Let $2\theta =u$.

So, $2d\theta =du$.

$$\begin{gathered} \int \cos 2\theta d\theta =\frac{1}{2}\int \cos udu \\ \begin{array}{rcl}& \Rightarrow & \int \cos 2\theta d\theta =\frac{\sin u}{2}+C\left[\mathbf{\because }\mathbf{\int }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{u}\mathbf{}\mathit{d}\mathit{u}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{u}\right]\\ & \Rightarrow & \int \cos 2\theta d\theta =\frac{\sin 2\theta }{2}+C\end{array} \end{gathered}$$

Hence, the integral of $\cos 2\theta$ is $\begin{array}{l}\frac{\sin 2\theta }{2}+C\end{array}$ where $C$ is a constant of integration.