How do you calculate the integral of polynomial functions?

To calculate the integral of polynomial functions, the power rule for integration is the appropriate method to use. The power rule states that for any polynomial term of the form ( ax^n ), where ( a ) is a coefficient and ( n ) is a positive integer, the integral can be computed by increasing the power by one and dividing by the new power. Specifically, the integral is given by:

[

\int ax^n , dx = \frac{a}{n+1} x^{n+1} + C

]

where ( C ) represents the constant of integration.

This rule is particularly useful for polynomial functions because they are comprised solely of terms of the form ( ax^n ). When integrating a polynomial, you apply the power rule to each term individually. For example, if you have a polynomial like ( 3x^3 + 2x^2 - x + 5 ), you would integrate each term separately:

[

\int (3x^3 + 2x^2 - x + 5) , dx = \left( \frac{3}{4}x^4 + \frac{2}{3}x^3 - \frac