Question

How to integrate $e^{2x}$?

Answer 1

Integrate the given expression :

Given, $\mathrm{I}=\int e^{2x}dx$

We will use the method of substitution to solve the integration.

Let $2x=z$

$$\begin{gathered} \Rightarrow 2dx=dz \\ \Rightarrow dx=\frac{1}{2}dz \end{gathered}$$

Therefore,

$$\begin{gathered} \mathrm{I}=\int e^{2x}dx \\ =\int \frac{1}{2}{e}^{z}dz \\ =\frac{1}{2}{e}^z+c[\because \int e^{u}du=e^u+c] \\ =\frac{1}{2}{e}^{2x}+c[\because z=2x] \end{gathered}$$

Hence, the value of integration of $\int e^{2x}dx$ is $\frac{1}{2}{e}^{2x}+c$.