Example Question - polynomial integration example

Integrating a Polynomial using Power Rule

You are asked to integrate a polynomial. The integral of a polynomial is found by integrating each term separately, according to the power rule for integration, \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) where \(C\) is the constant of integration. The integral of the polynomial \(8x^3 - x^2 + 5x - 1\) is calculated as follows: \[ \int (8x^3 - x^2 + 5x - 1)dx = \int 8x^3 dx - \int x^2 dx + \int 5x dx - \int 1dx \] Now integrate each term separately: \[ \int 8x^3 dx = 8 \times \frac{x^{3+1}}{3+1} = 2x^4 \] \[ \int x^2 dx = \frac{x^{2+1}}{2+1} = \frac{1}{3}x^3 \] \[ \int 5x dx = 5 \times \frac{x^{1+1}}{1+1} = \frac{5}{2}x^2 \] \[ \int 1dx = x \] So the integral is: \[ 2x^4 - \frac{1}{3}x^3 + \frac{5}{2}x^2 - x + C \] where \(C\) is the constant of integration.