1. **State the problem:** Find the indefinite integral $$\int \frac{e^{\sqrt{5x}}}{\sqrt{5x}} \, dx$$ including the constant of integration. 2. **Identify substitution:** Let $$u = \sqrt{5x} = (5x)^{1/2}$$. 3. **Differentiate substitution:** Then, $$u^2 = 5x \implies 2u \, du = 5 \, dx \implies dx = \frac{2u}{5} \, du$$. 4. **Rewrite the integral in terms of $$u$$:** $$\int \frac{e^u}{u} \cdot dx = \int \frac{e^u}{u} \cdot \frac{2u}{5} \, du = \int \frac{2}{5} e^u \, du$$. 5. **Simplify the integral:** $$\int \frac{2}{5} e^u \, du = \frac{2}{5} \int e^u \, du = \frac{2}{5} e^u + C$$. 6. **Back-substitute $$u$$:** $$\frac{2}{5} e^{\sqrt{5x}} + C$$. **Final answer:** $$\int \frac{e^{\sqrt{5x}}}{\sqrt{5x}} \, dx = \frac{2}{5} e^{\sqrt{5x}} + C$$