“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,

Using the 3rd law of logs,

We 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.

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?

So, where We need u from the substitution expression:

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.

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

Solutions: 1. Let

where To subs. back: So,

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

So,

SUMMARY The following results can be proved by trig substitutions:

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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:

So, We need u from the substitution expression: where

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

So,