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

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

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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:

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

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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

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Integration by Substitution Part 2 Solutions: Let 1.

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

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

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Integration by Substitution Part 2 So,

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Integration by Substitution Part 2 Show that x = sin 2 θ transforms x = sin 2 θ Using rule for brackets

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Integration by Substitution Part 2 Proven

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Integration by Substitution Part 2 This can be integrated using cos2 = 1–2sin 2 2sin 2 = 1–cos2

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Integration by Substitution Part 2 x = sin 2 θ So if x= ¼ sin 2 = ¼ sin = ½ = / 6 So if x= 0 sin 2 = 0 sin = 0 =

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Integration by Substitution Part 2 = / 6 =

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Integration by Substitution Part 2

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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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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:

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

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

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Integration by Substitution Part 2 So,

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