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Integrate the following polynomial: 2x3 - 5x2 + 4x - 1

May 21, 2026Mathematics

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Answer

\mathbf{\frac{1}{2}x^4 - \frac{5}{3}x^3 + 2x^2 - x + C}

3 steps

Apply the sum/difference rule and constant multiple rule for integration.
$\int (2x^3-5x^2+4x-1)\,dx = \int 2x^3\,dx - \int 5x^2\,dx + \int 4x\,dx - \int 1\,dx$ $= 2\int x^3\,dx - 5\int x^2\,dx + 4\int x\,dx - \int 1\,dx$
Apply the power rule for integration, $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ .
$2\left(\frac{x^{3+1}}{3+1}\right) - 5\left(\frac{x^{2+1}}{2+1}\right) + 4\left(\frac{x^{1+1}}{1+1}\right) - \left(\frac{x^{0+1}}{0+1}\right) + C$ $= 2\left(\frac{x^4}{4}\right) - 5\left(\frac{x^3}{3}\right) + 4\left(\frac{x^2}{2}\right) - x + C$
Simplify the expression.
$= \frac{2x^4}{4} - \frac{5x^3}{3} + \frac{4x^2}{2} - x + C$ $= \frac{1}{2}x^4 - \frac{5}{3}x^3 + 2x^2 - x + C$
The final answer is $\mathbf{\frac{1}{2}x^4 - \frac{5}{3}x^3 + 2x^2 - x + C}$ .

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