Integral of （4x^3-3x^2+2x+1） dx

1. Integrate term-by-term:

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

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

      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: $x^{4}$

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

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

      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: $x^{3}$

      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: $x^{2}$

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

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

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

2. Now simplify:

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

3. Add the constant of integration:

   $$x \left(x^{3} - x^{2} + x + 1\right)+ \mathrm{constant}$$

The answer is:

$$x \left(x^{3} - x^{2} + x + 1\right)+ \mathrm{constant}$$