1. Stating the problem: We need to find the integral of the function $x^3$ with respect to $x$. 2. Formula used: The integral of $x^n$ with respect to $x$ is given by $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1$ and $C$ is the constant of integration. 3. Applying the formula for $n=3$: $$\int x^3 \, dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C$$ 4. Explanation: We increase the power by 1 and divide by the new power. This is a basic rule for integrating power functions. Final answer: $$\int x^3 \, dx = \frac{x^4}{4} + C$$