Integral of 5x(3x-1)²dx dx

1. Rewrite the integrand:

   $$5 x \left(3 x - 1\right)^{2} = 45 x^{3} - 30 x^{2} + 5 x$$

2. Integrate term-by-term:

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

      $$\int 45 x^{3}\, dx = 45 \int x^{3}\, dx$$

      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: $\frac{45 x^{4}}{4}$

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

      $$\int \left(- 30 x^{2}\right)\, dx = - 30 \int x^{2}\, dx$$

      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: $- 10 x^{3}$

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

      $$\int 5 x\, dx = 5 \int x\, dx$$

      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: $\frac{5 x^{2}}{2}$

   The result is: $\frac{45 x^{4}}{4} - 10 x^{3} + \frac{5 x^{2}}{2}$

3. Now simplify:

   $$\frac{5 x^{2} \left(9 x^{2} - 8 x + 2\right)}{4}$$

4. Add the constant of integration:

   $$\frac{5 x^{2} \left(9 x^{2} - 8 x + 2\right)}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{5 x^{2} \left(9 x^{2} - 8 x + 2\right)}{4}+ \mathrm{constant}$$