Integral of 25x² dx

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

   $$\int 25 x^{2}\, dx = 25 \int x^{2}\, dx$$

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

      $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

   So, the result is: $\frac{25 x^{3}}{3}$

2. Add the constant of integration:

   $$\frac{25 x^{3}}{3}+ \mathrm{constant}$$

The answer is:

$$\frac{25 x^{3}}{3}+ \mathrm{constant}$$