1. **Problem:** Evaluate the integral $$\int \ln x \, dx$$ using integration by parts. 2. **Formula:** Integration by parts states: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = \ln x \quad \Rightarrow \quad du = \frac{1}{x} dx$$ $$dv = dx \quad \Rightarrow \quad v = x$$ 3. **Apply formula:** $$\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} dx = x \ln x - \int 1 \, dx$$ 4. **Simplify integral:** $$\int 1 \, dx = x$$ 5. **Final answer:** $$\int \ln x \, dx = x \ln x - x + C$$ This completes the solution for the first integral.