Integral of (x-3)(x²+3x+9)dx dx

1. Rewrite the integrand:

   $$\left(x - 3\right) \left(x^{2} + 3 x + 9\right) 1 = x^{3} - 27$$

2. Integrate term-by-term:

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

      $$\int x^{3}\, dx = \frac{x^{4}}{4}$$

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

      $$\int \left(-27\right)\, dx = - 27 x$$

   The result is: $\frac{x^{4}}{4} - 27 x$

3. Now simplify:

   $$\frac{x \left(x^{3} - 108\right)}{4}$$

4. Add the constant of integration:

   $$\frac{x \left(x^{3} - 108\right)}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{x \left(x^{3} - 108\right)}{4}+ \mathrm{constant}$$