1. **State the problem:** We need to evaluate the integral $$\int x \ln x \, dx$$. 2. **Choose a method:** Use integration by parts, where $$\int u \, dv = uv - \int v \, du$$. 3. **Assign parts:** Let $$u = \ln x$$ and $$dv = x \, dx$$. 4. **Compute derivatives and integrals:** - $$du = \frac{1}{x} \, dx$$ - $$v = \frac{x^2}{2}$$ 5. **Apply integration by parts formula:** $$\int x \ln x \, dx = \frac{x^2}{2} \ln x - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx = \frac{x^2}{2} \ln x - \frac{1}{2} \int x \, dx$$ 6. **Integrate remaining integral:** $$\int x \, dx = \frac{x^2}{2}$$ 7. **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:** $$\boxed{\int x \ln x \, dx = \frac{x^2}{2} \ln x - \frac{x^2}{4} + C}$$