Integral of x(x-1)^3 dx

1. Rewrite the integrand:

   $$x \left(x - 1\right)^{3} = x^{4} - 3 x^{3} + 3 x^{2} - x$$

2. Integrate term-by-term:

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

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

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

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

      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: $- \frac{3 x^{4}}{4}$

   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:

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

      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: $- \frac{x^{2}}{2}$

   The result is: $\frac{x^{5}}{5} - \frac{3 x^{4}}{4} + x^{3} - \frac{x^{2}}{2}$

3. Now simplify:

   $$\frac{x^{2} \left(4 x^{3} - 15 x^{2} + 20 x - 10\right)}{20}$$

4. Add the constant of integration:

   $$\frac{x^{2} \left(4 x^{3} - 15 x^{2} + 20 x - 10\right)}{20}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(4 x^{3} - 15 x^{2} + 20 x - 10\right)}{20}+ \mathrm{constant}$$