1. **State the problem:** We want to find the integral $$\int x \ln(x) \, dx$$. 2. **Formula and method:** We will use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ 3. **Choose parts:** Let $$u = \ln(x) \quad \Rightarrow \quad du = \frac{1}{x} dx$$ $$dv = x \, dx \quad \Rightarrow \quad v = \frac{x^2}{2}$$ 4. **Apply integration by parts:** $$\int x \ln(x) \, dx = \frac{x^2}{2} \ln(x) - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx = \frac{x^2}{2} \ln(x) - \frac{1}{2} \int x \, dx$$ 5. **Integrate remaining integral:** $$\int x \, dx = \frac{x^2}{2}$$ 6. **Substitute back:** $$\int x \ln(x) \, dx = \frac{x^2}{2} \ln(x) - \frac{1}{2} \cdot \frac{x^2}{2} + C = \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C$$ **Final answer:** $$\boxed{\int x \ln(x) \, dx = \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C}$$