Integral of x(3-x)⁷dx dx

1. Rewrite the integrand:

   $$x \left(3 - x\right)^{7} = - x^{8} + 21 x^{7} - 189 x^{6} + 945 x^{5} - 2835 x^{4} + 5103 x^{3} - 5103 x^{2} + 2187 x$$

2. Integrate term-by-term:

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

      $$\int \left(- x^{8}\right)\, dx = - \int x^{8}\, dx$$

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

         $$\int x^{8}\, dx = \frac{x^{9}}{9}$$

      So, the result is: $- \frac{x^{9}}{9}$

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

      $$\int 21 x^{7}\, dx = 21 \int x^{7}\, dx$$

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

         $$\int x^{7}\, dx = \frac{x^{8}}{8}$$

      So, the result is: $\frac{21 x^{8}}{8}$

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

      $$\int \left(- 189 x^{6}\right)\, dx = - 189 \int x^{6}\, dx$$

      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: $- 27 x^{7}$

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

      $$\int 945 x^{5}\, dx = 945 \int x^{5}\, dx$$

      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: $\frac{315 x^{6}}{2}$

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

      $$\int \left(- 2835 x^{4}\right)\, dx = - 2835 \int x^{4}\, dx$$

      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: $- 567 x^{5}$

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

      $$\int 5103 x^{3}\, dx = 5103 \int x^{3}\, dx$$

      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: $\frac{5103 x^{4}}{4}$

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

      $$\int \left(- 5103 x^{2}\right)\, dx = - 5103 \int x^{2}\, dx$$

      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: $- 1701 x^{3}$

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

      $$\int 2187 x\, dx = 2187 \int x\, dx$$

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

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

      So, the result is: $\frac{2187 x^{2}}{2}$

   The result is: $- \frac{x^{9}}{9} + \frac{21 x^{8}}{8} - 27 x^{7} + \frac{315 x^{6}}{2} - 567 x^{5} + \frac{5103 x^{4}}{4} - 1701 x^{3} + \frac{2187 x^{2}}{2}$

3. Now simplify:

   $$\frac{x^{2} \left(- 8 x^{7} + 189 x^{6} - 1944 x^{5} + 11340 x^{4} - 40824 x^{3} + 91854 x^{2} - 122472 x + 78732\right)}{72}$$

4. Add the constant of integration:

   $$\frac{x^{2} \left(- 8 x^{7} + 189 x^{6} - 1944 x^{5} + 11340 x^{4} - 40824 x^{3} + 91854 x^{2} - 122472 x + 78732\right)}{72}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(- 8 x^{7} + 189 x^{6} - 1944 x^{5} + 11340 x^{4} - 40824 x^{3} + 91854 x^{2} - 122472 x + 78732\right)}{72}+ \mathrm{constant}$$