Lesson Explainer: Indefinite Integrals: The Power Rule | Nagwa Lesson Explainer: Indefinite Integrals: The Power Rule | Nagwa

Lesson Explainer: Indefinite Integrals: The Power Rule Mathematics

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 𝑓 is another function 𝐹 whose derivative is equal to the original function 𝑓.

Definition: An Antiderivative of a Function

For any function 𝑓 defined on a subset 𝑈 and a differentiable function 𝐹𝑈, if we have 𝐹(𝑥)=𝑓(𝑥), then we say that 𝐹(𝑥) is an antiderivative of 𝑓(𝑥).

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

Definition: The Indefinite Integral

The indefinite integral of 𝑓(𝑥) with respect to 𝑥 can be written in terms of an antiderivative 𝐹(𝑥) as 𝑓(𝑥)𝑥=𝐹(𝑥)+,dC where C is also called the constant of integration.

Antiderivatives of 𝑓 always exist when 𝑓 is continuous, and there are infinitely many antiderivatives for 𝑓, obtained by adding the arbitrary constant C to 𝐹. This constant, also known as the constant of integration, is very important, as it produces a family of antiderivatives paramaterized by C. In other words, 𝐹(𝑥)+C is the most general function that has a derivative 𝑓(𝑥), for all C. For example, the derivative of 𝑥 is given by (𝑥)=1.

Thus, we can say that 𝑥 is an antiderivative of 1, but 𝑥+C is the most general antiderivative of 1, which means that 𝑥+1, 𝑥+7, 𝑥+2, 𝑥+𝜋, and so on are all also antiderivatives of 1. This is what we call the indefinite integral and is expressed as 1𝑥=𝑥+.dC

Similarly, the derivative of 𝑥 is 𝑥=2𝑥, which implies that the indefinite integral of 2𝑥 is 2𝑥𝑥=𝑥+.dC

Recall that the derivative satisfies the property (𝑎𝐹(𝑥))=𝑎𝐹(𝑥).

This means we can always take out a constant multiple outside the derivative. Thus, 𝑎𝐹(𝑥) is an antiderivative of 𝑎𝑓(𝑥), or 𝑎𝐹(𝑥)+C is the general antiderivative of 𝑎𝑓(𝑥), for all C, 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: (𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.dd

Using the indefinite integral of 2𝑥 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 2𝑥𝑥=2𝑥𝑥=𝑥+𝑥𝑥=𝑥2+,ddCdC where we note that we have kept the constant C the same since this is arbitrary and C2 is just another constant. In this explainer, we will be particularly interested in determining indefinite integrals of the form 𝑥𝑥d using the power rule of integration. We can determine this rule directly from the power rule of differentiation. Suppose 𝐹(𝑥)=𝑥, for 𝑝. The derivative of this function can be found by the power rule for differentiation as follows: (𝑥)=𝑝𝑥.

It will be useful to rewrite this as 1𝑝(𝑥)=1𝑝𝑝𝑥𝑥𝑝=𝑥, where we have divided by the constant 𝑝0, 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 𝑥, we want to determine the antiderivative). This means we want to find the most general function that differentiates to give 𝑥.

We have already shown that the derivative of 𝑥𝑝 is 𝑥, for 𝑝0. If we let 𝑝=𝑛+1, then we have 𝑥𝑛+1=𝑥,𝑛1.

Thus, 𝑥𝑛+1 is an antiderivative of 𝑥, provided 𝑛1. 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 𝑥, provided 𝑛1, as follows: 𝑥𝑥=𝑥𝑛+1+.dC

For example, using this power rule, we can determine the indefinite integral of 𝑥 as follows: 𝑥𝑥=𝑥2+1+=𝑥3+,dCC which can be verified directly by differentiating the right-hand side to obtain the integrand 𝑥.

In the first example, we will determine the indefinite integral of a function involving a positive integer power of 𝑥 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 𝑥𝑥d.

Answer

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

In order to determine the integral, we will make use of the following property of indefinite integrals: (𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.dd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

We can use the property to take out the factor of (1) outside the integral and determine the indefinite integral of 𝑥 using the power rule: 𝑥𝑥=𝑥𝑥=𝑥9+1+=𝑥10+.ddCC

Now, let’s consider an example where we will determine the indefinite integral of a function involving a negative integer power of 𝑥 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 27𝑥𝑥d.

Answer

In this example, we will determine the indefinite integral of a negative integer power of 𝑥, in particular the function 27𝑥.

In order to determine the integral, we will make use of the following property of indefinite integrals: (𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.dd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

We can use the property to take the factor of 27 outside the integral and determine the indefinite integral of 𝑥 using the power rule: 27𝑥𝑥=27𝑥𝑥=27𝑥9+1+=27𝑥8+=2𝑥56+=128𝑥+.ddCCCC

This result is valid for all 𝑥0, 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 𝑥 and not just integers, provided that power is not equal to 1. In the next example, we will determine the indefinite integral of a function involving a positive fractional power of 𝑥 by rewriting the radical in terms of a power of 𝑥 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 7𝑥𝑥d.

Answer

In this example, we will determine the indefinite integral of a positive fractional power of 𝑥, in particular the function 7𝑥.

Let’s first rewrite the integrand, by noting that 𝑥=𝑥, as 7𝑥=7𝑥=7𝑥.

In order to determine the integral, we will make use of the following property of indefinite integrals: (𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.dd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

We can use the property to take the factor of 7 outside the integral and determine the indefinite integral of 𝑥 using the power rule. Following this, we can rewrite the final answer back in terms of the square root: 7𝑥𝑥=7𝑥𝑥=7𝑥𝑥=7𝑥+1+=7𝑥+=72𝑥5+=145𝑥+=145𝑥+.dddCCCCC

This result is valid for all 𝑥0, 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 𝑥 by rewriting the radical in terms of a power of 𝑥 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 6𝑥𝑥d.

Answer

In this example, we will determine the indefinite integral of a negative fractional power of 𝑥, in particular the function 6𝑥.

First, let’s rewrite the integrand, by noting that 𝑥=𝑥, as 6𝑥=6𝑥=6𝑥.

In order to determine the integral, we will make use of the following property of indefinite integrals: (𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.dd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

We can use the property to take out the factor of 6 outside the integral and determine the indefinite integral of 𝑥 using the power rule: 6𝑥𝑥=6𝑥𝑥=6𝑥𝑥=6𝑥+1+=6𝑥+=68𝑥7+=487𝑥+.dddCCCC

This result is valid for all 𝑥>0, 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 (𝐹(𝑥)+𝐺(𝑥))=𝐹(𝑥)+𝐺(𝑥).

This also implies a similar rule for indefinite integrals: (𝑓(𝑥)+𝑔(𝑥))𝑥=𝑓(𝑥)𝑥+𝑔(𝑥)𝑥.ddd

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 +C 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 𝑓 and 𝑔 defined on a subset 𝑈, we have the linearity property (𝑎𝑓(𝑥)+𝑏𝑔(𝑥))𝑥=𝑎𝑓(𝑥)𝑥+𝑏𝑔(𝑥)𝑥,ddd for 𝑎,𝑏.

The power rule for integration along with this linearity property allow us to determine the indefinite integral involving sums of different powers of 𝑥 including polynomial, reciprocal, and radical functions. For example, we can determine the indefinite integral of the linear function 6𝑥+8 as follows: (6𝑥+8)𝑥=6𝑥𝑥+8𝑥=6𝑥𝑥+81𝑥=6𝑥1+1+8𝑥+=6𝑥2+8𝑥+=3𝑥+8𝑥+.dddddCCC

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 25𝑥65𝑥+36𝑥d.

Answer

In this example, we will determine the indefinite integral of the polynomial function 25𝑥65𝑥+36.

In order to determine the integral, we will make use of the following properties of indefinite integrals: (𝑓(𝑥)+𝑔(𝑥))𝑥=𝑓(𝑥)𝑥+𝑔(𝑥)𝑥,(𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.ddddd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

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: 25𝑥65𝑥+36𝑥=25𝑥𝑥+65𝑥𝑥+36𝑥=25𝑥𝑥65𝑥𝑥+361𝑥=25𝑥2+165𝑥1+1+36𝑥0+1=25𝑥365𝑥2+36𝑥=253𝑥652𝑥+36𝑥+.dddddddC

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, C.

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 (𝑥+4)𝑥4𝑥+16𝑥d.

Answer

In this example, we will determine the indefinite integral of the polynomial function (𝑥+4)𝑥4𝑥+16.

Let’s first simplify the integrand by distributing the brackets: (𝑥+4)𝑥4𝑥+16=𝑥4𝑥+16𝑥+4𝑥16𝑥+64=𝑥+64.

In order to determine the integral, we will make use of the following properties of indefinite integrals: (𝑓(𝑥)+𝑔(𝑥))𝑥=𝑓(𝑥)𝑥+𝑔(𝑥)𝑥,(𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.ddddd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

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: (𝑥+4)𝑥4𝑥+16𝑥=𝑥+64𝑥=𝑥𝑥+64𝑥=𝑥𝑥+641𝑥=𝑥3+1+64𝑥+=𝑥4+64𝑥+.ddddddCC

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, C.

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 𝑥225𝑥15𝑥d.

Answer

In this example, we will determine the indefinite integral of the rational function 𝑥225𝑥15.

Let’s first simplify the integrand by noting that the numerator is the difference of two squares and can be written as 𝑥225=(𝑥+15)(𝑥15); thus, the integrand becomes 𝑥225𝑥15=(𝑥+15)(𝑥15)(𝑥15)=𝑥+15, for 𝑥15. In order to determine the integral, we will make use of the following properties of indefinite integrals: (𝑓(𝑥)+𝑔(𝑥))𝑥=𝑓(𝑥)𝑥+𝑔(𝑥)𝑥,(𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.ddddd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

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: 𝑥225𝑥15𝑥=(𝑥+15)𝑥=𝑥𝑥+15𝑥=𝑥𝑥+151𝑥=𝑥1+1+15𝑥+=𝑥2+15𝑥+.ddddddCC

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, C.

This result is valid for all 𝑥15, 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 𝑥 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 8+89𝑥+75𝑥𝑥d.

Answer

In this example, we will determine the indefinite integral of the rational function 8+89𝑥+75𝑥.

Let’s first rewrite the integrand as powers of 𝑥: 8+89𝑥+75𝑥=8+89𝑥+75𝑥.

In order to determine the integral, we will make use of the following properties of indefinite integrals: (𝑓(𝑥)+𝑔(𝑥))𝑥=𝑓(𝑥)𝑥+𝑔(𝑥)𝑥,(𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.ddddd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

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: 8+89𝑥+75𝑥𝑥=8+89𝑥+75𝑥𝑥=8𝑥+89𝑥𝑥+75𝑥𝑥=81+89𝑥𝑥+75𝑥𝑥=8𝑥+89𝑥2+1+75𝑥6+1+=8𝑥+89𝑥1+75𝑥5+=8𝑥89𝑥725𝑥+𝐶.dddddddCC

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, C.

This result is valid for all 𝑥0, 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 𝑥+8+9𝑥𝑥d.

Answer

In this example, we will determine the indefinite integral of the function 𝑥+8+9𝑥.

Let’s first rewrite the integrand as powers of 𝑥 using 𝑥=𝑥: 𝑥+8+9𝑥=8𝑥+9𝑥, for 𝑥>0. In order to determine the integral, we will make use of the following properties of indefinite integrals: (𝑓(𝑥)+𝑔(𝑥))𝑥=𝑓(𝑥)𝑥+𝑔(𝑥)𝑥,(𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.ddddd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

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: 𝑥+8+9𝑥𝑥=8𝑥+9𝑥𝑥=8𝑥+𝑥𝑥+9𝑥𝑥=81𝑥𝑥𝑥+9𝑥𝑥=8𝑥𝑥+1+9𝑥2+1+=8𝑥𝑥+9𝑥1+=8𝑥23𝑥9𝑥+=8𝑥23𝑥9𝑥+.ddddddddCCCC

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, C.

This result is valid for all 𝑥>0, 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 36𝑥49𝑥(6𝑥+7)𝑥d.

Answer

In this example, we will determine the indefinite integral of the function 36𝑥49𝑥(6𝑥+7).

We note that the numerator in the integrand can be written as the difference of two squares, as 36𝑥49=(6𝑥)7=(6𝑥+7)(6𝑥7). Using this, we can simplify the integrand and rewrite the remaining terms as powers of 𝑥 using 𝑥=𝑥: 36𝑥49𝑥(6𝑥+7)=(6𝑥+7)(6𝑥7)𝑥(6𝑥+7)=6𝑥7𝑥=(6𝑥7)𝑥=6𝑥7𝑥, for 𝑥>0. In order to determine the integral, we will make use of the following properties of indefinite integrals: (𝑓(𝑥)+𝑔(𝑥))𝑥=𝑓(𝑥)𝑥+𝑔(𝑥)𝑥,(𝑎𝑓(𝑥))𝑥=𝑎𝑓(𝑥)𝑥.ddddd

We will also make use of the power rule: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

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: 36𝑥49𝑥(6𝑥+7)𝑥=6𝑥7𝑥𝑥=6𝑥𝑥+7𝑥𝑥=6𝑥𝑥7𝑥𝑥=6𝑥+17𝑥+1+=6𝑥+7𝑥+=68𝑥15+78𝑥7+=16𝑥58𝑥+.ddddddCCCC

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, C.

This result is valid for all 𝑥>0, 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 𝑥 including polynomial, reciprocal, and radical functions, we make use of the following:

  • The linearity property of integrals: (𝑎𝑓(𝑥)+𝑏𝑔(𝑥))𝑥=𝑎𝑓(𝑥)𝑥+𝑏𝑔(𝑥)𝑥,ddd for 𝑎,𝑏.
  • The power rule for integration: 𝑥𝑥=𝑥𝑛+1+,𝑛1.dC

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