Integral of (sinx-4x³)dx dx

1. Integrate term-by-term:

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

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

      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: $- x^{4}$

   1. The integral of sine is negative cosine:

      $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

   The result is: $- x^{4} - \cos{\left(x \right)}$

2. Add the constant of integration:

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

The answer is:

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