We are asked to compute the definite integral $\int_{0}^{4} f(x) dx$ for the function $f(x) = 4x^3 - 2x$.

1. Problem Description

We are asked to compute the definite integral $\int_{0}^{4} f(x) dx$ for the function $f(x) = 4x^3 - 2x$.

2. Solution Steps

First, we find the indefinite integral of $f(x)$.

$$\int f(x) dx = \int (4x^3 - 2x) dx$$

We can split the integral:

$$\int (4x^3 - 2x) dx = \int 4x^3 dx - \int 2x dx$$

Now we can integrate each term.

$$\int 4x^3 dx = 4 \int x^3 dx = 4 \cdot \frac{x^4}{4} = x^4$$

$$\int 2x dx = 2 \int x dx = 2 \cdot \frac{x^2}{2} = x^2$$

So the indefinite integral is

$$\int (4x^3 - 2x) dx = x^4 - x^2 + C$$

Now we can evaluate the definite integral:

$$\int_{0}^{4} (4x^3 - 2x) dx = \left[ x^4 - x^2 \right]_0^4$$

$$= (4^4 - 4^2) - (0^4 - 0^2)$$

$$= (256 - 16) - (0 - 0)$$

$$= 240$$

3. Final Answer

240