1. **Problem:** Use integration by parts to find the integral $$\int t \ln(t^2) \, dt$$. 2. **Formula:** Integration by parts formula is $$\int u \, dv = uv - \int v \, du$$. 3. **Choose parts:** Let $$u = \ln(t^2)$$ and $$dv = t \, dt$$. 4. **Compute derivatives and integrals:** - $$du = \frac{d}{dt} \ln(t^2) = \frac{2}{t} \, dt$$ - $$v = \int t \, dt = \frac{t^2}{2}$$ 5. **Apply formula:** $$\int t \ln(t^2) \, dt = \frac{t^2}{2} \ln(t^2) - \int \frac{t^2}{2} \cdot \frac{2}{t} \, dt = \frac{t^2}{2} \ln(t^2) - \int t \, dt$$ 6. **Simplify integral:** $$\int t \, dt = \frac{t^2}{2}$$ 7. **Final expression:** $$\int t \ln(t^2) \, dt = \frac{t^2}{2} \ln(t^2) - \frac{t^2}{2} + C$$ 8. **Simplify logarithm:** Since $$\ln(t^2) = 2 \ln|t|$$, $$\int t \ln(t^2) \, dt = \frac{t^2}{2} \cdot 2 \ln|t| - \frac{t^2}{2} + C = t^2 \ln|t| - \frac{t^2}{2} + C$$ **Answer:** $$\boxed{t^2 \ln|t| - \frac{t^2}{2} + C}$$