Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Integration - Reverse Chain Rule.
Concept and a summary of the steps.


 

On this page we discuss:
1. the concept of the Reverse Chain Rule technique;
2. what the questions requiring the technique look like;
3. How to apply the Reverse Chain Rule.

 

The concept:

The Reverse Chain Rule is a special technique for integrating a function having a particular structure. The function should consist of two components with one component being the derivative of the other. The technique combines these two parts by using another variable which makes the integration process more direct (and easier).

So if we have to integrate an expression of the form , we notice there are two parts:

In the integral , (x4 + 42) is the original or base compnent while (4x3) is the derivative component.

We then create another variable to represent the base part - so create f(x) = u = x4 + 42.

We now easily integrate with respect to the variable u. Then we replace the variable u with the original x term - i.e with f(x) - because the original question was expressed in terms of x.

 

What the questions look like.

There are many formats for functions which we need to integrate.

The Reverse Chain Rule is the most common format for the functions we encounter - especially at first.


The integrals requiring the Reverse Chain Rule technique are identified by having

one part of their expression being the derivative of the other part.


Examples of this format include:

The 2x is a derivative of the term inside the brackets. The numerator is the derivative of the expression in the denominator. The numerator is the derivative of the expression inside the brackets in the denominator. The index differentiates to give the denominator

 

How to apply the Reverse Chain Rule:

Integrations requiring this approach are integrated using the following steps:

Step 1: Identify the function with the highest power and let it equal a pronumeral of your choice - mostly we use the letter u
(for reasons I cannot explain - I use the letter which starts my name :-)).
Step 2: Differentiate (separating the derivative fraction by placing the dx - or equivalent expression - on the right side).
Step 3: Adjust the coefficients when necessary.
Step 4: Substitute, using your equations formed in Steps 1 and 3, for both components in the original equation.
Step 5: Do the integration.
Step 6: Substitute the expression for u (or whatever) back into the integrated equation.
NOTE: If you are evaluating a definite integral with limits given to you on the integral sign, ALWAYS finish Step 6 by coming back to the expression written in terms of the original value (x or t or whatever) before substituting.