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**“Teach A Level Maths” Vol. 2: A2 Core Modules**

27: Integration by Substitution Part 2 © Christine Crisp

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**A useful example of integration by substitution is to find**

We write Let So,

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Using the 3rd law of logs,

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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: Can you spot what to do next?

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So, where We need u from the substitution expression:

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Exercise 1. Find using the substitution 2. Show that using the substitution this is an example of the general result

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Solutions: 1. Let

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where To subs. back: So,

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2. Show that using the substitution Solution: So, Use the identity:

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So,

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**Show that x = sin2θ transforms**

Using rule for brackets

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Proven

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**This can be integrated using cos2q = 1–2sin2q**

2sin2q = 1–cos2q

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**x = sin2θ So if x= ¼ sin2q = ¼ sinq = ½ q = p/6 So if x= 0 sin2q = 0**

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q = p/6 q = 0

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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.

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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:

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So, We need u from the substitution expression: where

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2. Show that using the substitution Solution: So, Use the identity:

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So,

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