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IBMathsDPApplications & Interpretation (AI)HLRevision Notes5. CalculusFurther IntegrationIntegrating Special Functions

Integrating Special Functions (DP IB Applications & Interpretation (AI)): Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

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DP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HL

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Integrating Trig Functions

How do I integrate sin, cos and 1/cos2?

  • The antiderivatives for sine and cosine are

bold space bold integral bold sin bold italic x bold space bold d bold italic x bold equals bold minus bold cos bold italic x bold plus bold italic c

bold space bold integral bold cos bold italic x bold space bold d bold italic x bold equals bold sin bold italic x bold plus bold italic c

wherebold space bold italic c is the constant of integration

  • Also, from the derivative ofspace tan space x

bold space bold integral fraction numerator bold 1 over denominator bold cos to the power of bold 2 bold italic x end fraction bold space bold d bold italic x bold equals bold tan bold italic x bold plus bold italic c

Examiner Tips and Tricks

The three standard integrals above are all in the exam formula booklet.

  • For the linear functionbold space bold italic a bold italic x bold plus bold italic b, wherespace bold italic a andspace bold italic b are constants,

bold space bold integral bold sin bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold space bold d bold italic x bold equals bold minus bold 1 over bold italic a bold cos bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold plus bold italic c

bold space bold integral bold cos bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold space bold d bold italic x bold equals bold 1 over bold italic a bold sin bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold plus bold italic c

bold space bold integral fraction numerator bold 1 over denominator bold cos to the power of bold 2 bold left parenthesis bold italic a bold italic x bold italic plus bold italic b bold right parenthesis end fraction bold space bold d bold italic x bold equals bold 1 over bold italic a bold tan bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold plus bold italic c

  • For calculus with trigonometric functions angles must be measured in radians

    • Ensure you know how to change the angle mode on your GDC

Examiner Tips and Tricks

Make sure you have a copy of the formula booklet during revision but don't try to memorise everything in the formula booklet.

Instead, make sure you are familiar with the layout of the formula booklet. That way you’ll be able to quickly locate whatever you are after.

For formulae you think you have remembered, use the booklet to double check.

Worked Example

a)  Find, in the formspace straight F left parenthesis x right parenthesis plus c, an expression for each integral

  1. space integral cos x space straight d x

  2. space integral fraction numerator 1 over denominator cos squared open parentheses 3 x minus pi over 3 close parentheses end fraction space straight d x

5-4-1-ib-hl-ai-aa-extraaa-ai-we1a-soltn

b)  A curve has equationspace y equals integral 2 sin open parentheses 2 x plus pi over 6 close parentheses space straight d x.

The curve passes through the point with coordinatesspace open parentheses pi over 3 comma space square root of 3 close parentheses.

Find an expression forspace y.

5-4-1-ib-hl-ai-aa-extraaa-we1b-soltn-

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Integrating e^x & 1/x

How do I integrate exponentials and 1/x?

  • The antiderivatives involvingbold space bold e to the power of bold italic x andspace bold ln bold italic x are

bold space bold integral bold space bold e to the power of bold italic x bold space bold d bold italic x bold equals bold space bold e to the power of bold italic x bold plus bold italic c

where bold italic c is the constant of integration

Examiner Tips and Tricks

Both of the standard integrals above are in the exam formula booklet.

  • For the linear functionbold space bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis, wherespace bold italic a andspace bold italic b are constants,

 bold space bold integral bold e to the power of bold italic a bold italic x bold italic plus bold italic b end exponent bold space bold d bold italic x bold equals bold 1 over bold italic a bold e to the power of bold italic a bold italic x bold italic plus bold italic b end exponent bold plus bold italic c

  • It follows from the last result (by using the using the reverse chain rule) that

 

Examiner Tips and Tricks

With ln, it can sometimes be useful to write the constant of integration,space c, as a logarithm. I.e., by letting c equals ln k for some (positive) constant k.

Using the laws of logarithms, the answer can then be written as a single term. For example:

table row cell space integral 1 over x space straight d x end cell equals cell ln stretchy vertical line x stretchy vertical line plus c end cell row blank equals cell ln stretchy vertical line x stretchy vertical line plus ln k end cell row blank equals cell ln open parentheses k stretchy vertical line x stretchy vertical line close parentheses end cell end table

Worked Example

A curve has the gradient functionspace f to the power of apostrophe left parenthesis x right parenthesis equals fraction numerator 3 over denominator 3 x plus 2 end fraction plus straight e to the power of 4 minus x end exponent.


Given that the exact value ofspace f left parenthesis 1 right parenthesis isspace ln 10 minus straight e cubed, find an expression forspace f left parenthesis x right parenthesis.

NA5HYQ75_5-4-1-ib-sl-aa-only-we2-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.