1. **State the problem:** Calculate the definite integral $$\int_{-1}^1 (x^5 + 7x^4) \, dx$$. 2. **Recall the integral rules:** The integral of a sum is the sum of the integrals, and the integral of $$x^n$$ is $$\frac{x^{n+1}}{n+1}$$ for $$n \neq -1$$. 3. **Split the integral:** $$\int_{-1}^1 x^5 \, dx + \int_{-1}^1 7x^4 \, dx$$ 4. **Integrate each term:** - $$\int x^5 \, dx = \frac{x^6}{6} + C$$ - $$\int 7x^4 \, dx = 7 \cdot \frac{x^5}{5} + C = \frac{7x^5}{5} + C$$ 5. **Evaluate definite integrals:** $$\left[ \frac{x^6}{6} \right]_{-1}^1 + \left[ \frac{7x^5}{5} \right]_{-1}^1 = \left( \frac{1^6}{6} - \frac{(-1)^6}{6} \right) + \left( \frac{7 \cdot 1^5}{5} - \frac{7 \cdot (-1)^5}{5} \right)$$ 6. **Calculate values:** - $$\frac{1}{6} - \frac{1}{6} = 0$$ - $$\frac{7}{5} - \left(-\frac{7}{5}\right) = \frac{7}{5} + \frac{7}{5} = \frac{14}{5}$$ 7. **Sum results:** $$0 + \frac{14}{5} = \frac{14}{5}$$ **Final answer:** $$\int_{-1}^1 (x^5 + 7x^4) \, dx = \frac{14}{5}$$