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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusFurther IntegrationIntegrating with Reciprocal Trigonometric Functions

Integrating with Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA)): Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

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DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

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Integrating with reciprocal trigonometric functions

cosec (cosecant, csc), sec (secant) and cot (cotangent) are the reciprocal functions of sine, cosine and tangent respectively.

What are the antiderivatives involving reciprocal trigonometric functions?

  • integral sec squared x space d x equals tan x plus c

  • integral sec x space tan x space d x equals sec x plus c

  • integral cosec x space cot x space d x equals negative cosec x plus c

  • integral cosec squared x space d x equals negative cot x plus c

Examiner Tips and Tricks

These integral results are not in the formula booklet. However the results for the derivatives of tan x comma space sec x comma space cosec x and cot xare in the formula booklet, so those can be used 'the other way round' to deduce the respective antiderivatives.

  • Be careful with the negatives in the last two results

  • Remember the integration constant “+c” !

How do I integrate these if a linear function of x is involved?

  • All integration rules could apply alongside the results above

  • The use of reverse chain rule is particularly common

    • For linear functions the following results can be useful

      • integral sec squared open parentheses a x plus b close parentheses space straight d x equals 1 over a tan open parentheses a x italic plus b close parentheses plus c

      • integral sec open parentheses a x plus b close parentheses space tan open parentheses a x plus b close parentheses space d x equals 1 over a sec open parentheses a x plus b close parentheses plus c

      • integral cosec open parentheses a x plus b close parentheses space cot open parentheses a x plus b close parentheses space d x equals negative 1 over a cosec open parentheses a x plus b close parentheses plus c

      • integral cosec squared open parentheses a x plus b close parentheses space d x equals negative 1 over a cot open parentheses a x plus b close parentheses plus c

Examiner Tips and Tricks

These results are not in the formula booklet.

  • They are not essential to remember but can make problems easier

  • They can be deduced by spotting reverse chain rule

    • Remember to use 'adjust and compensate' for reverse chain rule when coefficients are involved

Worked Example

The graph of y equals f left parenthesis x right parenthesis where f left parenthesis x right parenthesis equals integral 2 sec squared 5 x space d x passes through the point open parentheses pi over 3 comma space 0 close parentheses.

Show that 5 y equals 2 open parentheses square root of 3 plus tan 5 x close parentheses.

5-9-1-ib-hl-aa-only-we1-soltn

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    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.