1. **State the problem:** We need to evaluate the integral $$\int x \ln(x^2) \, dx$$. 2. **Recall the formula and rules:** Use integration by parts formula: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = \ln(x^2), \quad dv = x \, dx$$ Then: $$du = \frac{d}{dx} \ln(x^2) \, dx = \frac{2}{x} \, dx = \frac{2}{x} dx, \quad v = \frac{x^2}{2}$$ 3. **Apply integration by parts:** $$\int x \ln(x^2) \, dx = \frac{x^2}{2} \ln(x^2) - \int \frac{x^2}{2} \cdot \frac{2}{x} \, dx = \frac{x^2}{2} \ln(x^2) - \int x \, dx$$ 4. **Simplify the remaining integral:** $$\int x \, dx = \frac{x^2}{2}$$ 5. **Write the final answer:** $$\int x \ln(x^2) \, dx = \frac{x^2}{2} \ln(x^2) - \frac{x^2}{2} + C$$ This is the evaluated integral with the constant of integration $C$.