Integral of s(3x²-2x+1)dx dx

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

   $$\int s \left(\left(3 x^{2} - 2 x\right) + 1\right)\, dx = s \int \left(\left(3 x^{2} - 2 x\right) + 1\right)\, dx$$

   1. Integrate term-by-term:

      1. Integrate term-by-term:

         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:

            $$\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$$

            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: $x^{3} - x^{2}$

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

         $$\int 1\, dx = x$$

      The result is: $x^{3} - x^{2} + x$

   So, the result is: $s \left(x^{3} - x^{2} + x\right)$

2. Now simplify:

   $$s x \left(x^{2} - x + 1\right)$$

3. Add the constant of integration:

   $$s x \left(x^{2} - x + 1\right)+ \mathrm{constant}$$

The answer is:

$$s x \left(x^{2} - x + 1\right)+ \mathrm{constant}$$