In this explainer, we will learn how to find the indefinite integrals of polynomials and general power functions using the power rule for integration.
Recall that for any function defined on a subset , we call a function the antiderivative of if for every . It follows that for any constant , the function is also an antiderivative of . This is expressed by the expression and we say that the expression is the indefinite integral of . This shorthand is particularly useful if the function is given by a formula involving an independent variable “”.
For example, the formula defines a function on .
Because , we have and . In other words, which is expressed in integral notation by
Recall the following properties of derivatives:
- for any functions .
- for any function and any constant .
- for ; in particular, .
Properties (1) and (2) mean that the sum of antiderivatives is the antiderivative of a sum of functions (additivity), and that the antiderivative of a constant multiple of a function is that same constant multiplied by the antiderivative (scalar multiples):
Of course, we must remember to add the constant to every indefinite integral.
Property (3) is especially useful when the functions are polynomials. It leads us to the formula (power functions): So, for example, because
Every polynomial is just a sum of constant multiples of powers of , so we combine these results in order to evaluate their indefinite integrals. For example:
Example 1: Integrating Monomials
The polynomial is . Its indefinite integral is, therefore,
Notice how we only consider the constant at the end.
Example 2: Integrating Polynomials
The integrand of the polynomial is . Its indefinite integral is, therefore,
If the polynomial is not written in the standard form (as a sum of powers of ), it is necessary to do this first before evaluating the integral by this method.
Example 3: Integrating Polynomials Involving Multiplying Out Brackets
First, simplify the integrand by expanding the brackets:
This integral can be performed directly: