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IBMathsDPAnalysis & Approaches (AA)SLRevision Notes5. CalculusFurther IntegrationIntegrating Special Functions

Integrating Special Functions (DP IB Analysis & Approaches (AA)): Revision Note

Roger B

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Roger B

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

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Integrating Trig Functions

How do I integrate sin and cos?

  • 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

    • These are given in the 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 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

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

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

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 3 sin stretchy left parenthesis 2 x plus straight pi over 3 stretchy right parenthesis space straight d x

iii) space integral left parenthesis 2 sin left parenthesis 4 x right parenthesis minus 3 cos left parenthesis 2 x right parenthesis right parenthesis space straight d x

 

5-4-1-ib-sl-aa-only-we1-soltn-a

b) The graph ofspace y equals straight F left parenthesis x right parenthesis plus c for question (a) part (ii) passes through the point with coordinatesspace stretchy left parenthesis straight pi over 3 comma space 5 over 2 stretchy right parenthesis.

Find the value ofspace c.

5-4-1-ib-sl-aa-only-we1-soltn-b

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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.

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

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    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.