DP IB Maths: AA HL

Revision Notes

5.3.1 Introduction to Integration

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Introduction to Integration

What is integration?

  • Integration is the opposite to differentiation
    • Integration is referred to as antidifferentiation
    • The result of integration is referred to as the antiderivative
  • Integration is the process of finding the expression of a function (antiderivative) from an expression of the derivative (gradient function)

What is the notation for integration?

  • An integral is normally written in the form

integral f left parenthesis x right parenthesis space straight d x 

    • the large operator integral means “integrate”
    • straight d x” indicates which variable to integrate with respect to
    • space f left parenthesis x right parenthesis is the function to be integrated (sometimes called the integrand)
  • The antiderivative is sometimes denoted by straight F left parenthesis x right parenthesis
    • there’s then no need to keep writing the whole integral; refer to it as straight F left parenthesis x right parenthesis
  • straight F left parenthesis x right parenthesis may also be called the indefinite integral ofspace f left parenthesis x right parenthesis

What is the constant of integration? 

  • Recall one of the special cases from Differentiating Powers of x
    • Ifspace f left parenthesis x right parenthesis equals a thenspace f apostrophe left parenthesis x right parenthesis equals 0
  • This means that integrating 0 will produce a constant term in the antiderivative
    • a zero term wouldn’t be written as part of a function
    • every function, when integrated, potentially has a constant term
  • This is called the constant of integration and is usually denoted by the letter c
    • it is often referred to as “plus c
  • Without more information it is impossible to deduce the value of this constant
    • there are endless antiderivatives, straight F left parenthesis x right parenthesis, for a functionspace f left parenthesis x right parenthesis

Integrating Powers of x

How do I integrate powers of x?

  • Powers ofspace x are integrated according to the following formulae:
    • Ifspace f left parenthesis x right parenthesis equals x to the power of n thenspace integral f left parenthesis x right parenthesis space straight d x equals fraction numerator x to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c wherespace n element of straight rational numbers comma space n not equal to negative 1 andspace c is the constant of integration

    • This is given in the formula booklet
  • If the power ofspace x is multiplied by a constant then the integral is also multiplied by that constant
    • Ifspace f left parenthesis x right parenthesis equals a x to the power of n thenspace integral f left parenthesis x right parenthesis space straight d x equals fraction numerator a x to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c wherespace n element of straight rational numbers comma space n not equal to negative 1 andspace a is a constant andspace c is the constant of integration
  • fraction numerator straight d y over denominator straight d x end fraction notation can still be used with integration
  • Note that the formulae above do not apply whenspace n equals negative 1 as this would lead to division by zero
  • Remember the special case:
    • space integral a space straight d x equals a x plus c
      • e.g. space integral 4 space straight d x equals 4 x plus c 
    • This allows constant terms to be integrated
  • Functions involving roots will need to be rewritten as fractional powers ofspace x first
    • eg. Ifspace f left parenthesis x right parenthesis equals 5 cube root of x then rewrite asspace f left parenthesis x right parenthesis equals 5 x to the power of 1 third end exponent and integrate
  • Functions involving fractions with denominators in terms ofbold space bold italic x will need to be rewritten as negative powers ofspace x first
    • e.g.  Ifspace f left parenthesis x right parenthesis equals 4 over x squared plus x squared then rewrite asspace f left parenthesis x right parenthesis equals 4 x to the power of negative 2 end exponent plus x squared and integrate    

  • The formulae for integrating powers ofspace x apply to all rational numbers so it is possible to integrate any expression that is a sum or difference of powers ofspace x
    • e.g.  Ifspace size 16px f size 16px left parenthesis size 16px x size 16px right parenthesis size 16px equals size 16px 8 size 16px x to the power of size 16px 3 size 16px minus size 16px 2 size 16px x size 16px plus size 16px 4 then
               
  • Products and quotients cannot be integrated this way so would need expanding/simplifying first
    • e.g.  Ifspace f begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals size 16px 8 size 16px x to the power of size 16px 2 size 16px left parenthesis size 16px 2 size 16px x size 16px minus size 16px 3 size 16px right parenthesis then

What might I be asked to do once I’ve found the anti-derivative (integrated)?

  • With more information the constant of integration,space c, can be found
  • The area under a curve can be found using integration

Exam Tip

  • You can speed up the process of integration in the exam by committing the pattern of basic integration to memory
    • In general you can think of it as 'raising the power by one and dividing by the new power'
    • Practice this lots before your exam so that it comes quickly and naturally when doing more complicated integration questions

Worked example

Given that

find an expression forspace y in terms ofspace x.

5-3-1-ib-sl-aa-version-we1-soltn

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Author: Paul

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.