1. **State the problem:** We want to evaluate the integral $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx$$. 2. **Use substitution:** Let $$u = \sqrt{x} = x^{1/2}$$. Then, $$x = u^2$$ and $$dx = 2u\,du$$. 3. **Rewrite the integral:** Substitute into the integral: $$\int \frac{e^u}{u} \cdot 2u\,du = \int 2e^u\,du$$. 4. **Simplify the integral:** The $$u$$ in denominator and numerator cancel out, leaving: $$2 \int e^u\,du$$. 5. **Integrate:** The integral of $$e^u$$ with respect to $$u$$ is $$e^u$$, so: $$2e^u + C$$. 6. **Back-substitute:** Replace $$u$$ with $$\sqrt{x}$$: $$2e^{\sqrt{x}} + C$$. **Final answer:** $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx = 2e^{\sqrt{x}} + C$$.