1. **State the problem:** Evaluate the integral $$\int (3x+5)^7 \, dx$$. 2. **Use substitution:** Let $$u = 3x + 5$$. 3. **Find differential:** Then $$du = 3 \, dx \implies dx = \frac{du}{3}$$. 4. **Rewrite integral:** Substitute into the integral: $$\int (3x+5)^7 \, dx = \int u^7 \cdot \frac{du}{3} = \frac{1}{3} \int u^7 \, du$$. 5. **Integrate:** Use the power rule for integration: $$\int u^7 \, du = \frac{u^{7+1}}{7+1} = \frac{u^8}{8}$$. 6. **Combine constants:** $$\frac{1}{3} \int u^7 \, du = \frac{1}{3} \cdot \frac{u^8}{8} = \frac{u^8}{24}$$. 7. **Substitute back:** Replace $$u$$ with $$3x+5$$: $$\int (3x+5)^7 \, dx = \frac{(3x+5)^8}{24} + C$$. **Final answer:** $$\boxed{\int (3x+5)^7 \, dx = \frac{(3x+5)^8}{24} + C}$$ **Note:** The step in the original solution that reintroduced $$3.dx$$ after substitution is incorrect and redundant. The correct substitution directly replaces $$dx$$ with $$\frac{du}{3}$$ once.