The chain rule for integrals is an integration rule related to the chain rule for derivatives. This rule is used for integrating functions of the form **f'( x)[f(x)]^{n}**.

Here, we will learn how to find integrals of functions using the chain rule for integrals. Then we will look at some examples where we will apply this rule.

## Formula for the chain of integration

The formula for the chain rule of integrals is as follows:

$$\int f'(x)[f(x)]^ndx=\frac{[f(x)]^{n+1}}{n+1}+c$$

We can understand this formula by considering the function $latex f(x)=(x^2+1)^4$. Using the chain rule of derivatives, we find that its derivative is:

$latex f'(x)=8x(x^2+1)^3$

This means that we can write as follows:

$$\int 8x(x^2+1)^3dx=(x^2+1)^4+c$$

Now, we can recognize that the integrand $latex 8x(x^2+1)^3$ is of the form $latex f'(x)[f(x)]^n$.

In this example we have $latex f(x)=x^2+1$, $latex f'(x)=2x$ and $latex n=3$.

To look at another example, let’s consider the integral $latex \int x(3x^2-2)^5dx$. In this case, we observe that the derivative of $latex (3x^2-2)$ is $latex 6x$. Moreover, we have the term $latex x$ outside the main function $latex (3x^2-2)^5$.

This means that we consider $latex (3x^2-2)^6$, which when differentiated gives us $latex 36x(3x^2-2)^5$. Then, we have:

$$\int x(3x^2-2)^5=\frac{1}{36}(3x^2-2)^6+c$$

## Solved exercises of the chain rule of integrals

**EXAMPLE 1**

**EXAMPLE 1**

Find the following integral:

$$\int (x-2)^2dx$$

##### Solution

To find the integral $latex \int (x-2)^2dx$, we can consider the function $latex f(x)=(x-2)^3$, which when derived gives us $latex f'(x)=3(x-2)^2$.

This means that the integral of $latex (x-2)^2$ is equal to:

$$\int (x-2)^2dx=\frac{(x-2)^3}{3}+c$$

**EXAMPLE **2

**EXAMPLE**Solve the following integral:

$$\int x(3x^2+6)^4dx$$

##### Solution

We can observe that the derivative of $latex 3x^2+6$ is $latex 6x$ and we have the term $latex x$ outside the main function $latex (3x^2+6)^4$.

This means we can consider that the derivative of $latex (3x^2+6)^5$ is $latex 30x(3x^2+6)^4$.

Then, we have the following integral:

$$\int x(3x^2+6)^4dx=\frac{(3x^2+6)^5}{30}+c$$

**EXAMPLE **3

**EXAMPLE**Find the following integral:

$$\int 4x^2(x^3-3)^5dx$$

##### Solution

To solve this integral, we observe that the derivative of $latex x^3-3$ is $latex 3x^2$.

Moreover, we have a term $latex x^2$ outside the main function $latex (x^3-3)^5$.

Then, we consider that the derivative of $latex (x^3-3)^6$ is $latex 18x^2(x^3-3)^5$. Thus, we have:

$$\int 4x^2(x^3-3)^5dx=\frac{4}{18}(x^3-3)^6+c$$

$$=\frac{2}{9}(x^3-3)^6+c$$

**EXAMPLE **4

**EXAMPLE**What is the result of the following integral?

$$\int (x+2)(x^2+4x-1)^3dx $$

##### Solution

We begin by noting that the derivative of $latex x^2+4x-1$ is $latex 2x+4=2(x+2)$.

Also, we see that we have the term $latex (x+2)$ outside the main function.

Then, considering that the derivative of $latex (x^2+4x-1)^4$ is

$$ 4(2x+4)(x^2+4x-1)^3=8(x+2)(x^2+4x-1)^3$$

Therefore, we have:

$$ \int (x+2)(x^2+4x-1)^3dx=\frac{1}{8}(x^2+4x-1)^4+c $$$$ =\frac{x(x^2-4)}{x^2-25} $$

**EXAMPLE **5

**EXAMPLE**Find the following integral:

$$ \int \frac{4x}{(3-x^2)^2}dx$$

##### Solution

In this case, we can start by writing the expression as follows to facilitate its resolution:

$$ \int \frac{4x}{(3-x^2)^2}dx=\int 4x (3-x^2)^{-2}$$

Now, we can observe that the derivative of $latex 3-x^2$ is $latex -2x$. Also, we have a term $latex x$ outside the main function.

Then, we consider that the derivative of $latex (3-x^2)^{-1}$ is $latex 2x(3-x^2)^{-2}$.

This means that this integral is solved as follows:

$$ \int \frac{4x}{(3-x^2)^2}dx=\frac{4}{2}(3-x^2)^{-1}+c $$

$$ =\frac{2}{3-x^2}+c $$

**EXAMPLE **6

**EXAMPLE**What is the result of the following integral?

$$ \int \frac{x}{\sqrt{x^2+3}}dx$$

##### Solution

We write the expression as follows:

$$ \int \frac{x}{\sqrt{x^2+3}}dx=\int x (x^2+3)^{-\frac{1}{2}}$$

Now, we see that the derivative of $latex x^2+3$ is $latex 2x$ and we have a term $latex x$ outside the main function.

Then, we can solve this by considering that the derivative of $latex (x^2+3)^{-{frac{1}{2}}}$ is:

$$2x\times \frac{1}{2} (x^2+3)^{-\frac{1}{2}}=x(x^2+3)^{-\frac{1}{2}}$$

This means that this integral is solved as follows:

$$ \int \frac{x}{\sqrt{x^2+3}}dx=(x^2+3)^{\frac{1}{2}}+c $$

$$ =\sqrt{x^2+3}+c $$

## Chain rule for integration – Practice problems

#### By solving the following integral, the result can be expressed as a fraction. What is the numerator? $$\int \frac{25x^4}{(3-x^5)^2}dx$$

Write the numerator in the input box.

## See also

Interested in learning more about integrals? You can take a look at these pages:

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