Question

What is the integral of $\tan 2x$?

Answer 1

Find the integral of $\tan 2x$.

Let $2x=u$.

So, $2dx=du$

Put the above value in$\begin{array}{rcl}& & \int \tan 2xdx\end{array}$:

$$\begin{gathered} \int \tan 2xdx=\frac{1}{2}\int \tan udu \\ \Rightarrow \int \tan 2xdx=\frac{1}{2}\ln |secu|+C\left\{\because \mathbf{\int }\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{u}\mathbf{}\mathit{d}\mathit{u}\mathbf{=}\mathit{l}\mathit{n}\mathbf{|}\mathit{s}\mathit{e}\mathit{c}\mathbf{}\mathit{u}\mathbf{|}\mathbf{+}\mathit{C}\right\} \\ \Rightarrow \int \tan 2xdx=\frac{1}{2}\ln |sec2x|+C\left\{\because u=2x\right\} \end{gathered}$$

Hence, the required integral is $\begin{array}{rcl}\int \tan 2xdx& =& \frac{1}{2}\ln |sec2x|+C\end{array}$ where $C$ is a constant.