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 an antiderivative, also known as an inverse derivative or primitive, of a function is another function whose derivative is equal to the original function .

### Definition: An Antiderivative of a Function

For any function defined on a subset and a differentiable function , if we have then we say that is an antiderivative of .

The antiderivative of a function is equivalent to the indefinite integral, which we define as follows.

### Definition: The Indefinite Integral

The indefinite integral of with respect to can be written in terms of an antiderivative as where is also called the constant of integration.

Antiderivatives of always exist when is continuous, and there are infinitely many antiderivatives for , obtained by adding the arbitrary constant to . This constant, also known as the constant of integration, is very important, as it produces a family of antiderivatives paramaterized by . In other words, is the most general function that has a derivative , for all . For example, the derivative of is given by

Thus, we can say that is an antiderivative of 1, but is the most general antiderivative of 1, which means that , , , , and so on are all also antiderivatives of 1. This is what we call the indefinite integral and is expressed as

Similarly, the derivative of is which implies that the indefinite integral of is

Recall that the derivative satisfies the property

This means we can always take out a constant multiple outside the derivative. Thus, is an antiderivative of , or is the general antiderivative of , for all , which is the same as the indefinite integral. In other words, if the derivative gets multiplied by a constant, the antiderivative also gets multiplied by the same constant and vice versa. This implies the following property of indefinite integrals:

Using the indefinite integral of as written above, we can take out the factor of 2 using this property and divide both sides of the expression by 2 to obtain where we note that we have kept the constant the same since this is arbitrary and is just another constant. In this explainer, we will be particularly interested in determining indefinite integrals of the form using the power rule of integration. We can determine this rule directly from the power rule of differentiation. Suppose , for . The derivative of this function can be found by the power rule for differentiation as follows:

It will be useful to rewrite this as where we have divided by the constant , as a constant multiple of a function does not affect the derivative or antiderivative. But what if we want to work in reverse? (i.e., given , we want to determine the antiderivative). This means we want to find the most general function that differentiates to give .

We have already shown that the derivative of is , for . If we let , then we have

Thus, is an antiderivative of , provided . We can express this in terms of an indefinite integral in the following definition.

### Rule: The Power Rule for Integration

The power rule for integration allows us to determine the indefinite integral of , provided , as follows:

For example, using this power rule, we can determine the indefinite integral of as follows: which can be verified directly by differentiating the right-hand side to obtain the integrand .

In the first example, we will determine the indefinite integral of a function involving a positive integer power of using the power rule along with the property that allows us to take a constant multiple outside the integral.

### Example 1: The Power Rule of Integration

Determine .

### Answer

In this example, we will determine the indefinite integral of a positive integer power of , in particular the function .

In order to determine the integral, we will make use of the following property of indefinite integrals:

We will also make use of the power rule:

We can use the property to take out the factor of outside the integral and determine the indefinite integral of using the power rule:

Now, let’s consider an example where we will determine the indefinite integral of a function involving a negative integer power of using the power rule along with the property that allows us to take a constant multiple outside the integral.

### Example 2: Finding the Integration of a Function Using the Power Rule for Integration with a Negative Exponent

Determine .

### Answer

In this example, we will determine the indefinite integral of a negative integer power of , in particular the function .

In order to determine the integral, we will make use of the following property of indefinite integrals:

We will also make use of the power rule:

We can use the property to take the factor of outside the integral and determine the indefinite integral of using the power rule:

This result is valid for all , since we require the integrand and integral to be continuous and well defined.

We can use the power rule to determine the indefinite integral of any power of and not just integers, provided that power is not equal to . In the next example, we will determine the indefinite integral of a function involving a positive fractional power of by rewriting the radical in terms of a power of and using the power rule along with the property that allows us to take a constant multiple outside the integral.

### Example 3: Finding the General Antiderivative of a Function Using the Power Rule of Integration with Fractional Exponents

Determine .

### Answer

In this example, we will determine the indefinite integral of a positive fractional power of , in particular the function .

Let’s first rewrite the integrand, by noting that , as

In order to determine the integral, we will make use of the following property of indefinite integrals:

We will also make use of the power rule:

We can use the property to take the factor of 7 outside the integral and determine the indefinite integral of using the power rule. Following this, we can rewrite the final answer back in terms of the square root:

This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the square root is only defined for nonnegative numbers.

Now, let’s consider an example where we will determine the indefinite integral of a function involving a negative fractional power of by rewriting the radical in terms of a power of and by using the power rule along with the property that allows us to take a constant multiple outside the integral.

### Example 4: Finding the Integration of a Function Using the Power Rule with Roots

Determine .

### Answer

In this example, we will determine the indefinite integral of a negative fractional power of , in particular the function .

First, let’s rewrite the integrand, by noting that , as

We will also make use of the power rule:

We can use the property to take out the factor of 6 outside the integral and determine the indefinite integral of using the power rule:

This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the 8th root is only defined for nonnegative numbers.

Recall that the derivative is a linear operation, as it satisfies

This also implies a similar rule for indefinite integrals:

Therefore, in order to determine the indefinite integral of a sum of functions, we just find an indefinite integral of each part separately and add the results together, not forgetting the at the end. Usually, we would obtain multiple constants for each part from the process of integration, but we can combine these into one constant. We can also combine this property with the one that allows us to take constants outside the integral.

### Property: Linearity Property of Integration

For any continuous functions and defined on a subset , we have the linearity property for .

The power rule for integration along with this linearity property allow us to determine the indefinite integral involving sums of different powers of including polynomial, reciprocal, and radical functions. For example, we can determine the indefinite integral of the linear function as follows:

In the next example, we will determine the indefinite integral of a polynomial function using the properties of linearity and the power rule for integrals.

### Example 5: Finding the Integration of a Polynomial Function Using the Power Rule

Determine .

### Answer

In this example, we will determine the indefinite integral of the polynomial function .

In order to determine the integral, we will make use of the following properties of indefinite integrals:

We will also make use of the power rule:

Using the first property, we can split the given integral into three parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:

Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .

Now, let’s consider an example where we will determine the indefinite integral of a polynomial function by distributing two brackets using the properties of linearity and the power rule for integrals.

### Example 6: Finding the Integration of a Polynomial Involving Multiplying Two Brackets and Applying the Power Rule

Determine .

### Answer

In this example, we will determine the indefinite integral of the polynomial function .

Let’s first simplify the integrand by distributing the brackets:

In order to determine the integral, we will make use of the following properties of indefinite integrals:

We will also make use of the power rule:

Using the first property, we can split the given integral into two parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:

Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .

In the next example, we will determine the indefinite integral of a rational function using factorization, the properties of linearity, and the power rule for integrals.

### Example 7: Finding the Integration of a Rational Function Using the Factorization of the Difference of Two Squares

Determine .

### Answer

In this example, we will determine the indefinite integral of the rational function .

Let’s first simplify the integrand by noting that the numerator is the difference of two squares and can be written as ; thus, the integrand becomes for . In order to determine the integral, we will make use of the following properties of indefinite integrals:

We will also make use of the power rule:

Using the first property we can split the given integral into two parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:

Note that we would obtain a constant of integration for each part from the process of integration, but we can combine these into one constant, .

This result is valid for all , since we require the integrand and integral to be continuous and well defined.

Now, let’s consider an example where we determine the indefinite integral of a rational function with negative powers of using the properties of linearity and the power rule for integrals.

### Example 8: Finding the Integration of a Function Using the Power Rule for Integration with a Negative Exponent

Determine .

### Answer

In this example, we will determine the indefinite integral of the rational function .

Let’s first rewrite the integrand as powers of :

In order to determine the integral, we will make use of the following properties of indefinite integrals:

We will also make use of the power rule:

Using the first property, we can split the given integral into three parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:

This result is valid for all , since we require the integrand and integral to be continuous and well defined.

In the next example, we will determine the indefinite integral of a function with roots and negative exponents using the properties of linearity and the power rule for integrals.

### Example 9: Finding the Integration of a Function Using the Power Rule for Integration with Roots and Negative Exponents

Determine .

### Answer

In this example, we will determine the indefinite integral of the function .

Let’s first rewrite the integrand as powers of using : for . In order to determine the integral, we will make use of the following properties of indefinite integrals:

We will also make use of the power rule:

Using the first property, we can split the given integral into three parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:

This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the square root is only defined for nonnegative numbers.

In the last example, we will determine the indefinite integral of a function involving fractional exponents by using factorization, the properties of linearity, and the power rule for integrals.

### Example 10: Finding the Integration of a Function Using Factorization

Determine .

### Answer

In this example, we will determine the indefinite integral of the function .

We note that the numerator in the integrand can be written as the difference of two squares, as . Using this, we can simplify the integrand and rewrite the remaining terms as powers of using : for . In order to determine the integral, we will make use of the following properties of indefinite integrals:

We will also make use of the power rule:

Using the first property, we can split the given integral into two parts. We can then use the second property to take the appropriate factors outside the integral and determine the indefinite integral of the different terms using the power rule:

This result is valid for all , since we require the integrand and integral to be continuous and well defined, and the 8th root is only defined for nonnegative numbers.

Let’s finish by considering the key points that we covered in this explainer.

### Key Points

In order to determine the indefinite integrals of functions involving different powers of including polynomial, reciprocal, and radical functions, we make use of the following:

- The linearity property of integrals: for .
- The power rule for integration: