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

Paul

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Reviewed by: Dan Finlay

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Integrating Powers of x

How do I integrate powers of x?

  • Powers ofspace x are integrated according to the following formula:

    • 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

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 rational power n (integers or fractions, positive or negative).

  • If the power ofspace x term 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

      • space a is a constant

      • andspace c is the constant of integration

  • 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

    • e.g. 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    

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.  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 to be expanded/simplified 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

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

find an expression forspace y in terms ofspace x.

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

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    Author
    Reviewer
    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.