1. **State the problem:** Evaluate the integral $$\int (3x+5)^7 \, dx$$. 2. **Formula and rule:** Use the substitution method for integrals of the form $$\int (ax+b)^n \, dx$$. 3. **Substitution:** Let $$u = 3x + 5$$, then $$\frac{du}{dx} = 3$$ or $$dx = \frac{du}{3}$$. 4. **Rewrite the integral:** $$\int (3x+5)^7 \, dx = \int u^7 \cdot \frac{du}{3} = \frac{1}{3} \int u^7 \, du$$. 5. **Integrate:** $$\frac{1}{3} \int u^7 \, du = \frac{1}{3} \cdot \frac{u^{8}}{8} + C = \frac{u^{8}}{24} + C$$. 6. **Back-substitute:** $$\frac{(3x+5)^8}{24} + C$$. **Final answer:** $$\int (3x+5)^7 \, dx = \frac{(3x+5)^8}{24} + C$$.