IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusTechniques & Applications of IntegrationIntegrating Trigonometric, Exponential & Reciprocal Functions

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

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Integrating Trig Functions

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

  • The antiderivatives 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

  • The derivatives ofspace sin space x comma space cos space x andspace tan space x are in the formula booklet

    • so these antiderivatives can be easily deduced

  • 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 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

  • The formula booklet can be used to find antiderivatives from the derivatives

    • Make sure you have the page with the section of standard derivatives open 

    • Use these backwards to find any antiderivatives you need

    • 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

  1. space integral cos space x space straight d x

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

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

b)       A curve has equationspace 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 coordinatesspace open parentheses straight 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 space 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

  • These are given in the 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 that

 

  • which can be deduced using Reverse Chain Rule

  • With ln, it can be useful to write the constant of integration,space c, as a logarithm

    • using the laws of logarithms, the answer can be written as a single term

    • wherespace k is a constant

    • This is similar to the special case of differentiatingspace ln space left parenthesis a x plus b right parenthesis whenspace b equals 0

Examiner Tips and Tricks

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

    • However, do be familiar with the layout of the formula booklet

      • You’ll be able to quickly locate whatever you are after

      • You do not want to be searching every line of every page!

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

Worked Example

A curve has the gradient functionspace f 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 the exact value ofspace f left parenthesis 1 right parenthesis isspace ln space 10 minus straight e cubed find an expression forspace f left parenthesis x right parenthesis.

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