Integral of x(x+1)(x+2) dx

1. Rewrite the integrand:

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

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 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: $x^{3}$

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

      $$\int 2 x\, dx = 2 \int x\, dx$$

      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: $x^{2}$

   The result is: $\frac{x^{4}}{4} + x^{3} + x^{2}$

3. Now simplify:

   $$x^{2} \left(\frac{x^{2}}{4} + x + 1\right)$$

4. Add the constant of integration:

   $$x^{2} \left(\frac{x^{2}}{4} + x + 1\right)+ \mathrm{constant}$$

The answer is: