1. **State the problem:** Find the antiderivative (indefinite integral) of the function $x^7$. 2. **Recall the formula:** The antiderivative of $x^n$ for $n \neq -1$ is given by $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $C$ is the constant of integration. 3. **Apply the formula:** Here, $n=7$, so $$\int x^7 \, dx = \frac{x^{7+1}}{7+1} + C = \frac{x^8}{8} + C$$ 4. **Explain:** We increase the exponent by 1 and divide by the new exponent. The constant $C$ accounts for any constant term that disappears when differentiating. **Final answer:** $$\frac{x^8}{8} + C$$