Integral of (8x^3+1) dx

1. Integrate term-by-term:

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

      $$\int 8 x^{3}\, dx = 8 \int x^{3}\, dx$$

      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: $2 x^{4}$

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

      $$\int 1\, dx = x$$

   The result is: $2 x^{4} + x$

2. Add the constant of integration:

   $$2 x^{4} + x+ \mathrm{constant}$$

The answer is:

$$2 x^{4} + x+ \mathrm{constant}$$