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.
Answer
\frac{1}{2} e^{2x} + C
Answer for screen readers
The final answer is \frac{1}{2} e^{2x} + C
Steps to Solve

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

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

Use the substitution method (if necessary)
To make integration simpler, we use substitution. Let $u = 2x$.

Find the differential of the substitution variable
Differentiate $u=2x$ with respect to $x$: $$du = 2 , dx$$ Hence, $dx = \frac{du}{2}$

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

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

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

Write the final answer
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
More Information
Integrals involving exponential functions are often easier to solve using substitution.
Tips
A common mistake is to forget to divide by the coefficient of $x$ inside the exponential function during integration.