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Integrating Powers of x (DP IB Applications & Interpretation (AI)): Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

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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

Examiner Tips and Tricks

Note that the formula can not be used if n equals negative 1, so you cannot integrate 1 over x this way.

Other than that, you can be asked to integrate powers of x for any integer power n space (i.e. space n equals 0 comma space plus-or-minus 1 comma space plus-or-minus 2 comma space plus-or-minus 3 comma space...).

  • 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

  • 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?

  • To integrate a sum or difference of power ofspace x terms, just integrate term by term

    • 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

Examiner Tips and Tricks

You can speed up the process of integration in the exam by committing the pattern of basic powers of x 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

    Author: Paul

    Expertise: Maths Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.

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