# Integrate e^(2x)

#### Understand the Problem

The question is asking to perform integration on the function e raised to the power of 2x. This indicates a mathematical process where we find the antiderivative of the function.

\frac{1}{2} e^{2x} + C

The final answer is \frac{1}{2} e^{2x} + C

#### Steps to Solve

1. Identify the function to integrate

The given function is $e^{2x}$. We need to integrate this with respect to $x$.

2. Set up the integral

Write the integral in standard form: $$\int e^{2x} , dx$$

3. Use the substitution method (if necessary)

To make integration simpler, we use substitution. Let $u = 2x$.

4. Find the differential of the substitution variable

Differentiate $u=2x$ with respect to $x$: $$du = 2 , dx$$ Hence, $dx = \frac{du}{2}$

5. Substitute into the integral

Replace $2x$ with $u$ and $dx$ with $\frac{du}{2}$: $$\int e^{u} , \frac{du}{2}$$ $$\frac{1}{2} \int e^{u} , du$$

6. Integrate the simplified integral

The integral of $e^{u}$ with respect to $u$ is $e^{u}$. So we get: $$\frac{1}{2} e^{u} + C$$

7. Substitute back the original variable

Replace $u$ back with $2x$: $$\frac{1}{2} e^{2x} + C$$

After substituting back, the integral of $e^{2x}$ with respect to $x$ is: $$\frac{1}{2} e^{2x} + C$$

The final answer is \frac{1}{2} e^{2x} + C

A common mistake is to forget to divide by the coefficient of $x$ inside the exponential function during integration.