DP IB Maths: AI SL

Revision Notes

5.2.2 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 straight 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
    • straight 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 of straight f left parenthesis x right parenthesis

What is the constant of integration? 

  • Recall one of the special cases from Differentiating Powers of x
    • If straight f left parenthesis x right parenthesis equals a then straight 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 function straight f left parenthesis x right parenthesis

Integrating Powers of x

How do I integrate powers of x? 

  • Powers of x are integrated according to the following formula:
    • If straight f stretchy left parenthesis x stretchy right parenthesis equals x to the power of n then integral straight 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 where n element of straight integer numbers comma double-struck    n not equal to negative 1 and c is the constant of integration

  • This is given in the formula booklet
  • If the power of is x multiplied by a constant then the integral is also multiplied by that constant
    • If straight f left parenthesis x right parenthesis equals a x to the power of n then integral straight 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 where n element of straight integer numbers comma double-struck    n not equal to negative 1,  a is a constant and 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 when n equals negative 1 as this would lead to division by zero
  • Don’t forget the special case:
    • integral a space straight d x equals a x plus c
      • e.g.  integral 4 space straight d x equals 4 x plus c
    • This allows constant terms to be integrated
  • Functions involving fractions with denominators in terms of x will need to be rewritten as negative powers of x first
    • e.g.  If straight f left parenthesis x right parenthesis equals 4 over x squared then rewrite as straight f left parenthesis x right parenthesis equals 4 x to the power of negative 2 end exponent and integrate

How do I integrate sums and differences of powers of x?

  • The formulae for integrating powers of x apply to all integers so it is possible to integrate any expression that is a sum or difference of powers of x
    • e.g.  If Error converting from MathML to accessible text. then integral straight f left parenthesis x right parenthesis space straight d x equals fraction numerator 8 x to the power of 3 plus 1 end exponent over denominator 3 plus 1 end fraction minus fraction numerator 2 x to the power of 1 plus 1 end exponent over denominator 1 plus 1 end fraction plus 4 x plus c equals 2 x to the power of 4 minus x squared plus 4 x plus c
  • Products and quotients cannot be integrated in this way so would need expanding/simplifying first
    • e.g. If  then integral straight f left parenthesis x right parenthesis space straight d x equals integral left parenthesis 16 x cubed minus 24 x squared right parenthesis space straight d x equals fraction numerator 16 x to the power of 4 over denominator 4 end fraction minus fraction numerator 24 x cubed over denominator 3 end fraction plus c equals 4 x to the power of 4 minus 8 x cubed plus c
       

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

fraction numerator straight d y over denominator straight d x end fraction equals 3 x to the power of 4 minus 2 x squared plus 3 minus 1 over x to the power of 4

find an expression for y in terms of x.

5-2-2-ib-sl-ai-aa-we-soltn

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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.