1. The problem is to find the indefinite integral of the function $$\frac{4}{4x} + x^3$$ with respect to $x$. 2. We can rewrite the function inside the integral as $$\frac{4}{4x} + x^3 = \frac{4}{4} \cdot \frac{1}{x} + x^3 = 1 \cdot \frac{1}{x} + x^3 = \frac{1}{x} + x^3$$. 3. The integral of a sum is the sum of the integrals, so: $$\int \left( \frac{1}{x} + x^3 \right) dx = \int \frac{1}{x} dx + \int x^3 dx$$. 4. Recall the integral formulas: - $$\int \frac{1}{x} dx = \ln|x| + C$$ - $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$. 5. Applying these formulas: $$\int \frac{1}{x} dx = \ln|x|$$ $$\int x^3 dx = \frac{x^{4}}{4}$$. 6. Therefore, the integral is: $$\int \left( \frac{1}{x} + x^3 \right) dx = \ln|x| + \frac{x^{4}}{4} + C$$. 7. This is the final answer, where $C$ is the constant of integration.