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

Calculus - Integration - Differentiate ... Hence find strategy.
Concept and a summary of the steps.


 

The concept:

The "Differentiate .. hence find" technique is used for integrating a function having a particular structure. Almost always, the questions are asked using those two words/phrases so they are easy to identify. Some clever questions - testing thinking skills separate the two words and/or use similar but not the same words.

The technique is based on the concept that on some occasions (not many) an integral can be made more direct if we differentiate a related function first. Then, using that derivative, the integral becomes manageable - sometimes a simple term and, at other times, maybe two (or more) terms can be produced.

For example, we may need to find . As a first step, the question asks us to find the derivative of .

As , we can integrate the terms on both sides of this equation and obtain the integral we need:

 

Naturally the question becomes easier when we are given the expression to differentiate.

 

The difference between the Differentiate ... hence find questions and the Reverse Chain Rule questions is essentially that:

 

How to apply the D-H-F strategy:

 

Step 1: Differentiate the term identified in the first part of the question - after the word differentiate :-) Differentiate
Step 2:

Write down what you have done using the prefix with the term you differentiated.

So the statements at the right say:

I differentiated (x2 - 5)3 and got
6x(x2 - 5)2
Step 3: Add integral signs and dx to every term.
Step 4: Integrate all terms you can - the one you cannot integrate will be the question!!!
Step 5: Rearrange and tidy up the terms to make your answer presentable.
NOTE: If you are evaluating a definite integral with limits given to you on the integral sign, ALWAYS finish Step 5 before substituting.