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):

and

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

Determine .

### Answer

The polynomial is . Its indefinite integral is, therefore,

Notice how we only consider the constant at the end.

### Example 2: Integrating Polynomials

Determine .

### Answer

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

Determine .

### Answer

First, simplify the integrand by expanding the brackets:

This integral can be performed directly: