1. **State the problem:** We need to find the integral $$\int x(2x-1)^7 \, dx$$. 2. **Formula and substitution:** Use substitution to simplify the integral. Let $$u = 2x - 1$$, then $$\frac{du}{dx} = 2$$ or $$dx = \frac{du}{2}$$. 3. Express $$x$$ in terms of $$u$$: $$u = 2x - 1 \Rightarrow x = \frac{u + 1}{2}$$. 4. Substitute into the integral: $$\int x(2x-1)^7 \, dx = \int \frac{u + 1}{2} u^7 \frac{du}{2} = \int \frac{u + 1}{2} u^7 \cdot \frac{1}{2} du = \int \frac{(u + 1) u^7}{4} du = \frac{1}{4} \int (u^{8} + u^{7}) du$$. 5. Integrate term by term: $$\frac{1}{4} \left( \frac{u^{9}}{9} + \frac{u^{8}}{8} \right) + C = \frac{1}{4} \left( \frac{u^{9}}{9} + \frac{u^{8}}{8} \right) + C$$. 6. Substitute back $$u = 2x - 1$$: $$\frac{1}{4} \left( \frac{(2x - 1)^{9}}{9} + \frac{(2x - 1)^{8}}{8} \right) + C = \frac{(2x - 1)^{9}}{36} + \frac{(2x - 1)^{8}}{32} + C$$. **Final answer:** $$\int x(2x-1)^7 \, dx = \frac{(2x - 1)^{9}}{36} + \frac{(2x - 1)^{8}}{32} + C$$.