Integrating Other Functions (Trig, ln & e etc) (CIE A Level Maths: Pure 3)

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Integrating Other Functions (Trig, ln & e etc)

Isn’t integration just the reverse of differentiation?

  • Yes, but remember “+c”, the constant of integration …
    • … unless finding a definite integral

  • Recognising common results helps to make integration easier

    (See Differentiating Other Functions)

  

  • You are given all the standard integrals in the formula booklet

 Notes diff_results, AS & A Level Maths revision notes

How do I integrate exponentials/ex? 

Notes ex, AS & A Level Maths revision notes

 

  • The gradient of ekx is kekx
    • ie y = ekx, dy/dx = kekx

  • The reverse applies when integrating
  • This is an example of reverse chain rule

 

Notes ekx, AS & A Level Maths revision notes

Integrating 1/x 

Notes ln, AS & A Level Maths revision notes

 

  • Remember 1/x = x-1
    • The method for integrating powers does not apply if the power is -1

 

Integrating sin and cos

Notes sin_cos, AS & A Level Maths revision notes

 

  • Note the minus in the integral of sin x
  • The integral of tan x is ln|sec x| + c

Integrating fraction numerator 1 over denominator 1 space plus space x squared end fraction and fraction numerator 1 over denominator a squared space plus space x squared end fraction

  • Recall that fraction numerator straight d over denominator straight d x end fraction left parenthesis tan to the power of negative 1 end exponent x right parenthesis space equals space fraction numerator 1 over denominator 1 space plus space x squared end fraction
  • Then integral fraction numerator 1 over denominator 1 space plus space x squared end fraction space straight d x space equals space tan to the power of negative 1 end exponent x space plus space c and integral fraction numerator 1 over denominator a squared space plus space x squared end fraction space straight d x space equals space 1 over a tan to the power of negative 1 end exponent open parentheses x over a close parentheses space plus space c

Examiner Tip

  • 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 locate quickly whatever you are after, and 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

Example soltn(a), AS & A Level Maths revision notesExample soltn(b), AS & A Level Maths revision notesExample soltn(c), AS & A Level Maths revision notes

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Lucy

Author: Lucy

Expertise: Head of STEM

Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.