Integrate the polynomial: integral (2x3-5x2+4x-1)\,dx

The power rule for integration states that $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$. We apply this rule to each term in the polynomial.

$$\int (2x^3-5x^2+4x-1)\,dx = \int 2x^3\,dx - \int 5x^2\,dx + \int 4x\,dx - \int 1\,dx$$

Integrate each term:

$$\int 2x^3\,dx = 2 \cdot \frac{x^{3+1}}{3+1} = 2 \cdot \frac{x^4}{4}$$ $$\int 5x^2\,dx = 5 \cdot \frac{x^{2+1}}{2+1} = 5 \cdot \frac{x^3}{3}$$ $$\int 4x\,dx = 4 \cdot \frac{x^{1+1}}{1+1} = 4 \cdot \frac{x^2}{2}$$ $$\int 1\,dx = x$$

$$\int (2x^3-5x^2+4x-1)\,dx = 2 \cdot \frac{x^4}{4} - 5 \cdot \frac{x^3}{3} + 4 \cdot \frac{x^2}{2} - x + C$$ $$= \frac{2x^4}{4} - \frac{5x^3}{3} + \frac{4x^2}{2} - x + C$$ $$= \frac{1}{2}x^4 - \frac{5}{3}x^3 + 2x^2 - x + C$$

The final answer is $\frac{1}{2}x^4 - \frac{5}{3}x^3 + 2x^2 - x + C$.