Evaluate the integral of 4x^4 - 2x^3 + x - 1 with respect to x.

To find the integral of the polynomial (4x^4 - 2x^3 + x - 1) with respect to (x), we use the power rule of integration. The power rule states that the integral of (x^n) is (\frac{x^{n+1}}{n+1}) plus a constant of integration (C).

Let’s compute the integral term by term:

1. For the term (4x^4):

[

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

]

2. For the term (-2x^3):

[

\int -2x^3 , dx = -2 \cdot \frac{x^{4}}{4} = -\frac{1}{2}x^4

]

3. For the term (x):

[

\int x , dx = \frac{x^{2}}{2}

]

4. For the constant term (-1):

[

\