O LevelFurther MathsCambridge (CIE)Additional MathsRevision NotesCalculusIntegrationIntegrating Trig Functions

Integrating Trig Functions (Cambridge (CIE) O Level Additional Maths): Revision Note

Exam code: 4037

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Integrating trig functions

How do I integrate sin, cos and sec^2?

  • The integrals for sine and cosine are

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

bold space bold integral bold cos bold space bold italic x bold space bold d bold italic x bold equals bold sin bold space 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 bold sec to the power of bold 2 bold space bold italic x bold space bold d bold italic x bold equals bold tan bold space bold italic x bold plus bold italic c

  • 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 space 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 space 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 space 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 space 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 sec to the power of bold 2 bold space 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 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

Examiner Tips and Tricks

  • Remember to add 'c', the constant of integration, for any indefinite integrals 

Worked Example

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

i. space integral cos space x space straight d x

ii. space integral sec squared space stretchy left parenthesis 3 x minus straight pi over 3 stretchy right parenthesis space straight d x

 

i.   This is a result you should be able to recall from memory.  

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

ii.   Use the standard result: integral sec to the power of 2 space end exponent open parentheses a x plus b close parentheses space d x space equals fraction numerator space 1 over denominator a end fraction tan open parentheses a x plus b close parentheses space plus space c space.  

  

b) A curve has the following equation:

space y equals integral 2 sin open parentheses 2 x plus straight pi over 6 close parentheses space straight d x

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

Find an expression for y.

 

Factor out the 2 in front of the sin first, and then use the standard result: integral sin open parentheses a x plus b close parentheses d x equals negative 1 over a cos open parentheses a x plus b close parentheses space plus c space.   

y equals 2 integral sin open parentheses 2 x plus straight pi over 6 close parentheses d x y equals 2 open parentheses negative 1 half cos open parentheses 2 x plus straight pi over 6 close parentheses plus c close parentheses  

When multiplying the arbitrary constant c, by 2, it is still just an arbitrary constant and so it is still just written as c.   

y equals minus cos open parentheses 2 x plus straight pi over 6 close parentheses plus c  

Substitute in the given coordinate, and evaluate, to find c.   

At x equals straight pi over 3 comma space y equals square root of 3

square root of 3 equals negative cos open parentheses fraction numerator 2 straight pi over denominator 3 end fraction plus straight pi over 6 close parentheses plus c square root of 3 equals negative cos open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses plus c square root of 3 equals fraction numerator square root of 3 over denominator 2 end fraction plus c c equals fraction numerator square root of 3 over denominator 2 end fraction

  Rewrite the full expression for y.  

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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.