Integral of (8^3+4x-7) dx

1. Integrate term-by-term:

   1. Integrate term-by-term:

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

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

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

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

         So, the result is: $2 x^{2}$

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

         $$\int 512\, dx = 512 x$$

      The result is: $2 x^{2} + 512 x$

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

      $$\int \left(-7\right)\, dx = - 7 x$$

   The result is: $2 x^{2} + 505 x$

2. Now simplify:

   $$x \left(2 x + 505\right)$$

3. Add the constant of integration:

   $$x \left(2 x + 505\right)+ \mathrm{constant}$$

The answer is:

$$x \left(2 x + 505\right)+ \mathrm{constant}$$