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Introduction to Integration (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Introduction to Integration

What is integration?

  • Integration is the inverse operation to differentiation

    • So if you differentiate a function to find its derivative

    • and then integrate that derivative

    • you should end up back at the original function

  • This can be written as  integral straight f to the power of apostrophe open parentheses x close parentheses space straight d x equals straight f open parentheses x close parentheses plus c

    • integral... space straight d x  is the integral with respect to bold italic x of "..."

    • c is the constant of integration

      • A derivative gives the rate of change of a function

        • but it doesn't give the starting point for any change

      • The constant of integration represents this 'starting point'

      • See the 'Constants of Integration' revision note for more info

  • This type of integral is known as an indefinite integral

    • The answer to an indefinite integral is another function

      • There are also definite integrals

        • The answer to a definite integral is a number

      • See the 'Calculating Areas' revision note for more info on definite integrals

  • Usually you will integrate using standard formulae

    • See the 'Integrating Basic Functions' revision note for these formulae

Examiner Tips and Tricks

  • Remember that integration and differentiation are inverse operations

    • So if you differentiate your answer to an indefinite integral

      • you should end up back at the function you were integrating

    • Use this to check your answers on the exam!

  • Don't forget the constant of integration (plus c) when finding an indefinite integral

    • Leaving it out can lose marks

Worked Example

(a) Show that the derivative of  x cubed plus 1 over x  is  3 x squared minus 1 over x squared.


This is a standard 'powers of x'derivative
Just remember to rewrite 1 over x as a power using laws of indices

straight f open parentheses x close parentheses equals x cubed plus x to the power of negative 1 end exponent

table row cell straight f to the power of apostrophe open parentheses x close parentheses end cell equals cell 3 x to the power of 3 minus 1 end exponent plus open parentheses open parentheses negative 1 close parentheses x to the power of negative 1 minus 1 end exponent close parentheses end cell row blank equals cell 3 x squared minus x to the power of negative 2 end exponent end cell end table


Now just use laws of indices again to write the answer in the requested form

table row cell bold f to the power of bold apostrophe stretchy left parenthesis x stretchy right parenthesis end cell bold equals cell bold 3 bold italic x to the power of bold 2 bold minus bold 1 over bold x to the power of bold 2 end cell end table


(b) Hence write down the answer to the indefinite integral  integral open parentheses 3 x squared minus 1 over x squared close parentheses space straight d x.

We can use  integral straight f to the power of apostrophe open parentheses x close parentheses space d x equals straight f open parentheses x close parentheses plus c to write down the answer

This is because integration and differentiation are inverse operations

Just don't forget the constant of integration  plus c

bold integral stretchy left parenthesis 3 x squared minus 1 over x squared stretchy right parenthesis bold space bold d bold italic x bold equals bold italic x to the power of bold 3 bold plus bold 1 over bold x bold plus bold italic c

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.