1. The problem is to verify the integral of $x \ln x$ using integration by parts. 2. Recall the integration by parts formula: $$\int u\,dv = uv - \int v\,du$$ 3. Choose $u = \ln x$ and $dv = x\,dx$. 4. Then, $du = \frac{1}{x} dx$ and $v = \frac{x^2}{2}$. 5. Apply the formula: $$\int x \ln x\, dx = \frac{x^2}{2} \ln x - \int \frac{x^2}{2} \cdot \frac{1}{x} dx = \frac{x^2}{2} \ln x - \int \frac{x}{2} dx$$ 6. Simplify the integral: $$\int \frac{x}{2} dx = \frac{1}{2} \int x dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4}$$ 7. Substitute back: $$\int x \ln x\, dx = \frac{x^2}{2} \ln x - \frac{x^2}{4} + C$$ 8. This matches the given result, confirming the solution is correct.