1. **State the problem:** We need to find the integral of the function $5e^{2x}$ with respect to $x$. 2. **Recall the formula:** The integral of $e^{ax}$ with respect to $x$ 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, the function is $5e^{2x}$, so $a=2$ and there is a constant multiplier 5 outside the exponential. 4. **Integrate step-by-step:** $$\int 5e^{2x} \, dx = 5 \int e^{2x} \, dx = 5 \cdot \frac{1}{2} e^{2x} + C$$ 5. **Simplify the expression:** $$= \frac{5}{2} e^{2x} + C$$ 6. **Final answer:** $$\int 5e^{2x} \, dx = \frac{5}{2} e^{2x} + C$$ This means the antiderivative of $5e^{2x}$ is $\frac{5}{2} e^{2x}$ plus a constant of integration.