1. **State the problem:** Evaluate the integral $$\int x^4 (x^5 - 7)^{31} \, dx$$. 2. **Identify the substitution:** Notice the inner function inside the power is $x^5 - 7$. Let’s set $$u = x^5 - 7$$. 3. **Differentiate $u$:** $$\frac{du}{dx} = 5x^4 \implies du = 5x^4 \, dx$$. 4. **Rewrite the integral in terms of $u$:** We have $$x^4 \, dx = \frac{du}{5}$$, so the integral becomes $$\int x^4 (x^5 - 7)^{31} \, dx = \int (u)^{31} \frac{du}{5} = \frac{1}{5} \int u^{31} \, du$$. 5. **Integrate with respect to $u$:** Using the power rule for integration, $$\int u^{31} \, du = \frac{u^{32}}{32} + C$$. 6. **Substitute back to $x$:** $$\frac{1}{5} \cdot \frac{u^{32}}{32} + C = \frac{1}{160} (x^5 - 7)^{32} + C$$. **Final answer:** $$\int x^4 (x^5 - 7)^{31} \, dx = \frac{(x^5 - 7)^{32}}{160} + C$$.