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

2. Module C3 Module C4 AQA Edexcel MEI/OCR OCR
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3. Integration by substitution can be used for a variety of integrals: some compound functions, some products and some quotients.
Sometimes we have a choice of method.

4. We have already met this type of integral.
See if you can do it.
Solution:
Reversing the chain rule gives
We’ll use this example to illustrate integration by substitution but if you got it right you can continue to use the reverse chain rule ( also called inspection ).

5. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
Find dx by treating like a fraction

6. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx

7. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx

8. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx
Integrate:
Replace u:

9. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx
Integrate:
Replace u:

10. e.g. 1
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx
Integrate:
Replace u:

11. Students studying the OCR specification should skip the next slide.

12. e.g. 2
This is a product but we can’t use integration by parts. Why not?
ANS: is a compound function with inner function non-linear. We can’t integrate it.
If we chose to integrate x instead, at the next stage we would have a more complicated integral than the one we started with.
We substitute as before, but using the inner function of the 2nd factor in the product.

13. Cancel the extra x e.g. 2 Define u as the inner function: Let
Differentiate:
Substitute for the inner function and dx
Cancel the extra x
If x won’t cancel we will have to make an extra substitution. We’ll do an example later.

14. So,
where
Integrate:
Substitute back:

15. SUMMARY
Substitution can be used for a variety of integrals
e.gs.
Method:
Define u as the inner function
Differentiate the substitution expression and rearrange to find dx
Substitute for the inner function and dx
If there’s an extra x, cancel it
If x won’t cancel, rearrange the substitution expression to find x and substitute for it
Integrate
Substitute back

16. Exercises
Use substitution to integrate the following. (Where possible, you could also use a 2nd method.) 1. 2. 3.

17. Solutions: 1.
Let
So,

18. Solutions: 2.
Let
So,

19. 3.
Solutions:
Let
So,

20. Definite integration
We work in exactly the same way BUT we must also substitute for the limits, since they are values of x and we are changing the variable to u.
A definite integral gives a value so we never return to x.

21. e.g. 1
Let
Limits:
So,

22. So,

23. e.g. 2
Let
Limits:
So,

24. So,
You will often see this written as
where
We leave answers in the exact form.

25. Exercises 2. 1.
Give exact answers.

26. 1.
Solutions:
Let
Limits: So

27. 2.
Let
Limits:
So,
We can use the log laws to simplify this.

28. In the next examples, the extra x doesn’t conveniently cancel so we need to substitute for it.

29. The extra x doesn’t cancel so we must substitute for it.
e.g. 3
Another product. This time we could use integration by parts but instead we’ll use substitution.
Define u as the inner function:
Let
Differentiate:
Substitute for the inner function and dx
The extra x doesn’t cancel so we must substitute for it.
Using
So,

30. So,
( where )
Can you spot the important difference between these?
Ans: We can easily multiply out the brackets in the 2nd
Integrate:
Replace u:

31. e.g. 4
Let
Tip: Don’t be tempted to substitute for the extra x . . .
until you’ve checked to see if it cancels.

32. x doesn’t cancel so now substitute:
e.g. 4
Let
So,
x doesn’t cancel so now substitute:
A multiplying constant . . .
can be taken outside the integral.

33. So,
where

34. So,
where

35. So,
where
Remove the brackets and substitute for u:

36. Exercise
Use substitution to integrate the following:

37. Solution:
Let
So,

39. 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.

40. SUMMARY
e.gs.
Differentiate the substitution expression and rearrange to find dx
Method:
Substitute for the inner function and dx
Define u as the inner function
If there’s an extra x, cancel it
If x won’t cancel, rearrange the substitution expression to find x and substitute for it
Substitution can be used for a variety of integrals
Integrate
Substitute back

41. Let
e.g. 1
Differentiate:
Method:
We must substitute for x and dx.
Substitute for the inner function
Define u as the inner function
and dx
Integrate:
Replace u:

42. Sometimes x won’t cancel and we have to make an extra substitution
Let
e.g. 2
Differentiate:
Substitute for the inner function and dx
Define u as the inner function:
Cancel the extra x
Sometimes x won’t cancel and we have to make an extra substitution

43. where
Integrate:
Replace u:
So,

44. The extra x doesn’t cancel so we must substitute for it.
Let
e.g. 3
Differentiate:
Substitute for the inner function and dx
Define u as the inner function:
The extra x doesn’t cancel so we must substitute for it.
Using
So,

45. Now we can easily multiply out the brackets
( where )
Integrate:
Replace u:
So,

46. Definite integration
We work in exactly the same way BUT we must also substitute for the limits, since they are values of x and we are changing the variable to u.
A definite integral gives a value so we never return to x.

47. e.g. 1
Let
Limits:
So,

48. where
So,
You will often see this written as