Integral of x²(5-x)⁴dx dx

1. Rewrite the integrand:

   $$x^{2} \left(5 - x\right)^{4} = x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 625 x^{2}$$

2. Integrate term-by-term:

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

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

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

      $$\int \left(- 20 x^{5}\right)\, dx = - 20 \int x^{5}\, dx$$

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

         $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

      So, the result is: $- \frac{10 x^{6}}{3}$

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

      $$\int 150 x^{4}\, dx = 150 \int x^{4}\, dx$$

      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: $30 x^{5}$

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

      $$\int \left(- 500 x^{3}\right)\, dx = - 500 \int x^{3}\, dx$$

      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: $- 125 x^{4}$

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

      $$\int 625 x^{2}\, dx = 625 \int x^{2}\, dx$$

      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: $\frac{625 x^{3}}{3}$

   The result is: $\frac{x^{7}}{7} - \frac{10 x^{6}}{3} + 30 x^{5} - 125 x^{4} + \frac{625 x^{3}}{3}$

3. Now simplify:

   $$\frac{x^{3} \left(3 x^{4} - 70 x^{3} + 630 x^{2} - 2625 x + 4375\right)}{21}$$

4. Add the constant of integration:

   $$\frac{x^{3} \left(3 x^{4} - 70 x^{3} + 630 x^{2} - 2625 x + 4375\right)}{21}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{3} \left(3 x^{4} - 70 x^{3} + 630 x^{2} - 2625 x + 4375\right)}{21}+ \mathrm{constant}$$