What is the integral of the polynomial expression x^5 - 2x^3 + x - 5?

To find the integral of the polynomial expression (x^5 - 2x^3 + x - 5), we apply the basic rules of integration for polynomials. The integral of (x^n) is given by (\frac{x^{n+1}}{n+1}) plus a constant of integration (C).

Applying this rule to each term of the polynomial:

1. For the term (x^5), the integral is (\frac{x^{6}}{6}).

2. For the term (-2x^3), the integral is (-2 \cdot \frac{x^{4}}{4} = -\frac{1}{2}x^{4}).

3. For the term (x), the integral is (\frac{x^{2}}{2}).

4. For the constant term (-5), the integral is (-5x).

Now, combining all these results gives:

[

\frac{x^{6}}{6} - \frac{1}{2}x^{4} + \frac{x^{2}}{2} - 5x + C.

]

When rearranging this,