1. We are asked to evaluate the integral $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx$$. 2. To solve this, use the substitution method. Let $$t = \sqrt{x}$$, so $$x = t^2$$ and $$dx = 2t \, dt$$. 3. Substitute into the integral: $$\int \frac{e^t}{t} \cdot 2t \, dt = \int 2 e^t \, dt$$. 4. Simplify the integral: $$\int 2 e^t \, dt = 2 \int e^t \, dt$$. 5. The integral of $$e^t$$ is $$e^t$$, so: $$2 e^t + C$$. 6. Substitute back $$t = \sqrt{x}$$: $$2 e^{\sqrt{x}} + C$$. Therefore, the solution to the integral is: $$\boxed{2 e^{\sqrt{x}} + C}$$.