Integral of 4-4x^3 dx

1. Integrate term-by-term:

   1. The integral of a constant is the constant times the variable of integration:

      $$\int 4\, dx = 4 x$$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- 4 x^{3}\right)\, dx = - \int 4 x^{3}\, dx$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 4 x^{3}\, dx = 4 \int x^{3}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

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

         So, the result is: $x^{4}$

      So, the result is: $- x^{4}$

   The result is: $- x^{4} + 4 x$

2. Now simplify:

   $$x \left(4 - x^{3}\right)$$

3. Add the constant of integration:

   $$x \left(4 - x^{3}\right)+ \mathrm{constant}$$

The answer is:

$$x \left(4 - x^{3}\right)+ \mathrm{constant}$$