Question

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

Answer 1

Find the integration of $\tan 2x$.

$$\begin{gathered} \mathrm{Let},2x=u \\ \Rightarrow 2dx=du\left[\because Differenciationw.r.tox\right] \end{gathered}$$

Put the above values in $\int \left(\tan 2x\right)dx$ ,to find the required integration as follow:

$\begin{array}{rcl}\int \tan 2xdx& =& \frac{1}{2}\int \tan udu\\ & \Rightarrow & \int \tan 2xdx=\frac{1}{2}\ln \left|secu\right|+C\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{\int }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{u}\mathbf{=}\mathbf{l}\mathbf{n}\mathbf{|}\mathbf{s}\mathbf{e}\mathbf{c}\mathbf{}\mathbf{u}\mathbf{|}\mathbf{+}\mathbf{C}\mathbf{,}\mathbf{}\mathbf{C}\mathbf{}\mathbf{i}\mathbf{s}\mathbf{}\mathbf{a}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{t}\mathbf{]}\end{array}$

$\begin{array}{rcl}& \Rightarrow & \int \tan 2xdx=\frac{1}{2}\ln \left|sec2x\right|+C\mathbf{[}\mathbf{\because }\mathit{u}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{]}\end{array}$

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