1. **State the problem:** Evaluate the integral $$\int x^8 \ln(x) \, dx$$ using integration by parts. 2. **Recall the integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$ 3. **Choose parts:** Let $$u = \ln(x)$$ and $$dv = x^8 \, dx$$. 4. **Compute derivatives and integrals:** $$du = \frac{1}{x} \, dx$$ $$v = \int x^8 \, dx = \frac{x^9}{9}$$ 5. **Substitute into the formula:** $$\int x^8 \ln(x) \, dx = \ln(x) \cdot \frac{x^9}{9} - \int \frac{x^9}{9} \cdot \frac{1}{x} \, dx$$ 6. **Simplify the integral:** $$= \frac{x^9}{9} \ln(x) - \frac{1}{9} \int x^{9-1} \, dx = \frac{x^9}{9} \ln(x) - \frac{1}{9} \int x^8 \, dx$$ 7. **Integrate:** $$\int x^8 \, dx = \frac{x^9}{9}$$ 8. **Final expression:** $$\int x^8 \ln(x) \, dx = \frac{x^9}{9} \ln(x) - \frac{1}{9} \cdot \frac{x^9}{9} + C = \frac{x^9}{9} \ln(x) - \frac{x^9}{81} + C$$ 9. **Answer:** $$\boxed{\int x^8 \ln(x) \, dx = \frac{x^9}{9} \ln(x) - \frac{x^9}{81} + C}$$