Integral of (4x^6) dx

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

   $$\int 4 x^{6}\, dx = 4 \int x^{6}\, dx$$

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

      $$\int x^{6}\, dx = \frac{x^{7}}{7}$$

   So, the result is: $\frac{4 x^{7}}{7}$

2. Add the constant of integration:

   $$\frac{4 x^{7}}{7}+ \mathrm{constant}$$

The answer is:

$$\frac{4 x^{7}}{7}+ \mathrm{constant}$$