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

2. Module C3 Module C4 AQA Edexcel OCR
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3. A useful example of integration by substitution is to find
We write
Let
So,

4. Using the 3rd law of logs,

5. 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.

6. 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?

7. So,
where
We need u from the substitution expression:

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

10. Solutions: 1.
Let

11. where
To subs. back:
So,

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

13. So,

14. SUMMARY
The following results can be proved by trig substitutions:

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. 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. So,
We need u from the substitution expression:
where

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

20. So,