Integral of 25x^(4)-3x^(2)-8 dx

1. Integrate term-by-term:

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

      $$\int 25 x^{4}\, dx = 25 \int x^{4}\, dx$$

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

         $$\int x^{4}\, dx = \frac{x^{5}}{5}$$

      So, the result is: $5 x^{5}$

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

      $$\int \left(- 3 x^{2}\right)\, dx = - \int 3 x^{2}\, dx$$

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

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

         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: $x^{3}$

      So, the result is: $- x^{3}$

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

      $$\int \left(\left(-1\right) 8\right)\, dx = - 8 x$$

   The result is: $5 x^{5} - x^{3} - 8 x$

2. Now simplify:

   $$x \left(5 x^{4} - x^{2} - 8\right)$$

3. Add the constant of integration:

   $$x \left(5 x^{4} - x^{2} - 8\right)+ \mathrm{constant}$$

The answer is:

$$x \left(5 x^{4} - x^{2} - 8\right)+ \mathrm{constant}$$