Integration by the Reverse Chain Rule
Neil Trivedi
Teacher
Integration by the Reverse Chain Rule
As a reminder, the chain rule is used to differentiate composite functions of the form , where is what we call the “angle” of our function.
We’d first differentiate the angle and write it at the front. Then, we identify what type of function is and differentiate it according to our standard rules to get . Then, the angle inside doesn’t change. It gives us:
Angle stays the same
Differentiated angle
With integration by the reverse chain rule, we simply reverse this process.
Angle stays the same
Differentiated angle
We identify the angle , the type of function , and the differentiated angle . The differentiated angle may be multiplied by a constant in the integrand. If so, we will adjust for that constant later.
If the function is not written clearly enough for us to identify these components, we may need to rewrite it so we can.
General Steps on Integrating by the Reverse Chain Rule
1) Identify the type of function, the angle, and the potential differentiated angle. If these are difficult to see, rewrite the function first.
2) We use integration by recognition where we make a guess (or guesses if there’s more than one term) as to what may differentiate, via the chain rule, to give us the function we’re integrating.
3) Differentiate the guess using the chain rule to check if we get what we are trying to integrate.
4) If the derivative is a constant multiple of our function, adjust by multiplying the guess by , or we can see it as multiplying by the constant we want over the constant we have. Note that we may only multiply or divide by constants, not variables.
5) Don’t forget to add a at the end (for indefinite integrals).
Example 1:
Integrate with respect to .
Step 1: Identify the type of function, the angle, and the potential differentiated angle.
We can see that we will be integrating a power function, where the angle is what’s inside the bracket. Here, the angle is , while the potential differentiated angle is .
Angle
Potential differentiated anglePower function
Step 2: Make a guess as to what may differentiate to give .
We will guess . With power functions, the power goes up by and in the process of integration, the differentiated angle will disappear.
Step 3: Differentiate the guess using the chain rule.
Here, we are differentiating a power function, so the angle is what is inside the bracket. Here, it is , which differentiates to give . Then, for the power function, we bring down the power, knock off the power, then keep the angle the same.
Angle stays the same
Differentiated angleDifferentiated power function
Step 4: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
No answer provided.
Example 2:
Integrate with respect to .
Step 1: Identify the type of function, the angle, and the potential differentiated angle.
We can see that we will be integrating an exponential function, where the angle is the power. Here, the angle is , while the potential differentiated angle is .
Angle
Exponential functionPotential differentiated angle
Step 2: Make a guess as to what may differentiate to give .
We will guess . In the process of integration, the differentiated angle will disappear and that we know the function of stays the same in differentiation and the power stays the same.
Step 3: Differentiate the guess using the chain rule.
Here, we are differentiating an exponential function, so the angle is what is in the power. Here, it is , which differentiates to give . Then, function of stays the same in differentiation as well as its power.
Angle stays the same
Differentiated angle
Step 3: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by .
No answer provided.
Example 3:
Integrate with respect to .
Step 1: Identify the type of function, the angle, and potential differentiated angle.
When we see fractions, we are immediately thinking about . This integral can be rewritten as
The angle has been reciprocated.
Potential differentiated angleAngle
We can identify that this will integrate to give a function of because we have a differentiated angle, which is , multiplied by the reciprocal of the angle (with the angle being ).
Step 2: Make a guess as to what may differentiate to give .
We will guess . In the process of integration, the differentiated angle will disappear.
Step 3: Differentiate the guess using the chain rule.
Here, we are differentiating a logarithmic function, so the angle is the argument (what we’re taking of). The angle is , which differentiates to . Then, when we differentiate the function of , the angle is reciprocated.
When is differentiated, the angle is reciprocated.
Differentiated angle
Step 4: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .
No answer provided.
Example 4:
Integrate with respect to .
Step 1: Rewrite the function so that we can identify the angle and potential differentiated angle.
Firstly, the fact that we see a power of gives away that it is a power function which we can then rewrite as . Initially, we notice that the angle is , while the potential differentiated angle is . However, when we differentiate , we would get .
So, the potential differentiated angle differs by a factor of compared to the actual differentiated angle. As a reminder, we may only multiply or divide by constants, not by variables, and is a variable. Therefore, we must rewrite the function so that the potential differentiated angle differs only by a constant factor.
We need the differentiated angle to be in the form , so we factorise out .
Angle
Power FunctionPotential differentiated angle
Step 2: Make a guess as to what may differentiate to give .
We will guess . With power functions, the power goes up by and the differentiated angle will disappear in the process of integration.
Step 3: Differentiate the guess using the chain rule.
Here, we are differentiating a power function, so the angle is what is inside the bracket. Here, it is , which differentiates to give . Then, for the power function, we bring down the power, knock off the power, then keep the angle the same.
Angle stays the same
Differentiated angleDifferentiated power function
Step 4: Adjust the constants.
Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have which will be .
No answer provided.
Challenging Question