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© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 27: Integration by Substitution Part 2 Part 2.

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Presentation on theme: "© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 27: Integration by Substitution Part 2 Part 2."— Presentation transcript:

1 © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 27: Integration by Substitution Part 2 Part 2

2 Integration by Substitution Part 2 "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" Module C3 AQA Edexcel OCR Module C4

3 Integration by Substitution Part 2 A useful example of integration by substitution is to find We write Let So,

4 Integration by Substitution Part 2 Using the 3 rd law of logs,

5 Integration by Substitution Part 2 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. We have already shown that Since indefinite integration is the reverse of differentiation,

6 Integration by Substitution Part 2 e.g. 1 Let N.B. Instead of defining u as a function of x we have defined x as a function of u. So, Can you spot what to do next? Use the identity:

7 Integration by Substitution Part 2 So, We need u from the substitution expression: where

8 Integration by Substitution Part 2 We 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.

9 Integration by Substitution Part 2 Exercise 1. Find using the substitution 2. Show that using the substitution this is an example of the general result

10 Integration by Substitution Part 2 Solutions: Let 1.

11 Integration by Substitution Part 2 where To subs. back: So,

12 Integration by Substitution Part 2 2. Show that using the substitution Solution: So, Use the identity:

13 Integration by Substitution Part 2 So,

14 Integration by Substitution Part 2 SUMMARY The following results can be proved by trig substitutions:

15 Integration by Substitution Part 2

16 The 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.

17 Integration by Substitution Part 2 e.g. 1 Let N.B. Instead of defining u as a function of x we have defined x as a function of u. So, Use the identity:

18 Integration by Substitution Part 2 So, We need u from the substitution expression: where

19 Integration by Substitution Part 2 2. Show that using the substitution Solution: So, Use the identity:

20 Integration by Substitution Part 2 So,


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