1. **State the problem:** We need to find the integral of the function $e^{2x}$ with respect to $x$. 2. **Formula and rules:** The integral of an exponential function $e^{ax}$ is given by $$\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C$$ where $a$ is a constant and $C$ is the constant of integration. 3. **Apply the formula:** Here, $a = 2$, so $$\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C$$ 4. **Explanation:** We divide by the coefficient of $x$ in the exponent to reverse the chain rule from differentiation. 5. **Final answer:** $$\boxed{\frac{1}{2} e^{2x} + C}$$