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

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

3. Using the 3rd law of logs,

4. 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?

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

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

7. Solutions: 1.
Let

8. where
To subs. back:
So,

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

10. So,

11. Show that x = sin2θ transforms
Using rule for brackets

12. Proven

13. This can be integrated using cos2q = 1–2sin2q
2sin2q = 1–cos2q

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

15. q = p/6
q = 0

17. 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.

18. 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:

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

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

21. So,