1. **State the problem:** We want to evaluate the integral $$\int \frac{\ln x}{\sqrt{x}} \, dx$$. 2. **Rewrite the integral:** Recall that $$\sqrt{x} = x^{1/2}$$, so the integral becomes $$\int \ln x \cdot x^{-1/2} \, dx$$. 3. **Use integration by parts:** The formula for integration by parts is $$\int u \, dv = uv - \int v \, du$$. 4. **Choose parts:** Let $$u = \ln x$$ and $$dv = x^{-1/2} dx$$. 5. **Compute derivatives and integrals:** - $$du = \frac{1}{x} dx$$ - $$v = \int x^{-1/2} dx = \int x^{-\frac{1}{2}} dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{1/2}$$ 6. **Apply integration by parts:** $$\int \ln x \cdot x^{-1/2} dx = u v - \int v \, du = 2 x^{1/2} \ln x - \int 2 x^{1/2} \cdot \frac{1}{x} dx$$ 7. **Simplify the integral:** $$\int 2 x^{1/2} \cdot \frac{1}{x} dx = 2 \int x^{1/2 - 1} dx = 2 \int x^{-1/2} dx$$ 8. **Evaluate the remaining integral:** $$2 \int x^{-1/2} dx = 2 \cdot 2 x^{1/2} = 4 x^{1/2}$$ 9. **Combine all parts:** $$\int \frac{\ln x}{\sqrt{x}} dx = 2 x^{1/2} \ln x - 4 x^{1/2} + C$$ 10. **Final answer:** $$\boxed{2 \sqrt{x} \ln x - 4 \sqrt{x} + C}$$