1. **Problem:** Find the integral $\int e^{2x} \, dx$. 2. **Formula and rule:** The integral of $e^{ax}$ with respect to $x$ is $\frac{1}{a} e^{ax} + C$, where $a$ is a constant and $C$ is the constant of integration. 3. **Solution:** Here, $a=2$, so $$\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C.$$ 4. **Explanation:** We use the rule for integrating exponential functions. Since the exponent is $2x$, we divide by 2 to compensate for the chain rule in differentiation. **Final answer:** $$\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C.$$