1. **State the problem:** Evaluate the integral $$\int 5^{\frac{x}{2}} \, dx$$. 2. **Recall the formula:** The integral of an exponential function with base $a$ is given by $$\int a^{u} \, du = \frac{a^{u}}{\ln(a)} + C$$ where $a > 0$, $a \neq 1$, and $u$ is a function of $x$. 3. **Identify the inner function:** Here, $u = \frac{x}{2}$, so $du = \frac{1}{2} dx$ or equivalently $dx = 2 \, du$. 4. **Rewrite the integral:** $$\int 5^{\frac{x}{2}} \, dx = \int 5^{u} \, (2 \, du) = 2 \int 5^{u} \, du$$ 5. **Integrate:** $$2 \int 5^{u} \, du = 2 \cdot \frac{5^{u}}{\ln(5)} + C = \frac{2}{\ln(5)} 5^{\frac{x}{2}} + C$$ 6. **Final answer:** $$\int 5^{\frac{x}{2}} \, dx = \frac{2}{\ln(5)} 5^{\frac{x}{2}} + C$$