2Module C3 Module C4 AQA Edexcel OCR "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
3A useful example of integration by substitution is to find We writeLetSo,
5We have already shown that Since indefinite integration is the reverse of differentiation,We can show this result directly by using a trig substitution.The following examples and exercises are difficult. You are unlikely to be asked to do them in an exam but you will find it useful to follow the method.
6e.g. 1LetN.B. Instead of defining u as a function of x we have defined x as a function of u.So,Use the identity:Can you spot what to do next?
7So,whereWe need u from the substitution expression:
8We can get a more general result by a similar method: Check that this is in your formula book. You can then quote it without proof and use it for any value of a.However, you may like to try using substitution for examples in the next exercise.
9Exercise1. Findusing the substitution2. Show thatusing the substitutionthis is an example of the general result
16The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
17e.g. 1LetN.B. Instead of defining u as a function of x we have defined x as a function of u.So,Use the identity:
18So,We need u from the substitution expression:where
192. Show thatusing the substitutionSolution:So,Use the identity: