1. “Teach A Level Maths” Vol. 2: A2 Core Modules
23: Integrating some Compound Functions
© Christine Crisp

2. Module C3 Module C4 AQA Edexcel MEI/OCR
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3. Before we try to integrate compound functions, we need to be able to recognise them, and know the rule for differentiating them.
where , the inner function. If
We saw that in words this says:
differentiate the inner function
multiply by the derivative of the outer function
we get
e.g. For

4. Since indefinite integration is the reverse of differentiation, we get3 3
So,
If we divide C by 3, we get another constant, say C1, but we usually just write C.
The rule is:
integrate the outer function

5. Since indefinite integration is the reverse of differentiation, we get
So,
The rule is:
integrate the outer function
divide by the derivative of the inner function

6. Since indefinite integration is the reverse of differentiation, we get
So,
The rule is:
integrate the outer function
divide by the derivative of the inner function

7. Since indefinite integration is the reverse of differentiation, we get
So,
The rule is:
integrate the outer function
divide by the derivative of the inner function
Tip: We can check the answer by differentiating it. We should get the function we wanted to integrate.

8. However, we can’t integrate all compound functions in this way.
Let’s try the rule on another example:
THIS IS WRONG !
e.g.
integrate the outer function
divide by the derivative of the inner function

9. However, we can’t integrate all compound functions in this way.
Let’s try the rule on another example:
THIS IS WRONG !
e.g.
The rule has given us a quotient, which, if we differentiate it, gives:
. . . nothing like the function we wanted to integrate.

10. ? What is the important difference between and
When we differentiate the inner function of the 1st example, we get 3, a constant.
Dividing by the 3 does NOT give a quotient of the form ( since v is a function of x ).
The 2nd example gives 2x,which is a function of x.

11. ? What is the important difference between and
When we differentiate the inner function of the 1st example, we get 3, a constant.
Dividing by the 3 does NOT give a quotient of the form ( since v is a function of x ).
The 2nd example gives 2x,which is a function of x.
So, the important difference is that the 1st example has an inner function that is linear; it differentiates to a constant. The 2nd is non-linear.

12. SUMMARY
The rule for integrating a compound function ( a function of a function ) is:
integrate the outer function
divide by the derivative of the inner function
provided that
the inner function is linear
Add C
There is NO single rule for integration if the inner function is non-linear.

13. Exercises
Without working them out, decide which of the following can be integrated using the rule we have found in this section. 1. 2. 3. 4. 5. 6. 7. 8.
We’ll now practise some integrals like 1, 2, 3, 6 and 8.

14. e.g. 1.
e.g. 2.
Integrate the outer function

15. e.g. 1.
e.g. 2.
Integrate the outer function

16. e.g. 1.
e.g. 2.
Integrate the outer function
Divide by the derivative of the inner function -3.
If we write
we have a clumsy “piled up” fraction so we put the (-3 ) beside the 5.

17. e.g. 1.
e.g. 2.
Integrate the outer function
Divide by the derivative of the inner function -3.

18. e.g. 3.
This is related to
Integrate the outer function:

19. e.g. 3.
This is related to
Integrate the outer function:
Divide by the derivative of the inner function

20. e.g. 3.
This is related to
Integrate the outer function:
Divide by the derivative of the inner function

21. e.g. 3.
This is related to
Integrate the outer function:
Divide by the derivative of the inner function

22. Exercises
Find 1. 2. 3. 4. 5.
Solutions: 1. 2.

23. 3. 4. 5.

25. 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.

26. The rule for integrating a compound function ( a function of a function ) is:
SUMMARY
integrate the outer function
divide by the derivative of the inner function
provided that
the inner function is linear
There is NO single rule for integration if the inner function is non-linear.
Add C

27. Only the functions marked with a tick can be integrated by the rule.1. 2. 3. 4. 5. 6. 7. 8.

28. Integrate the outer function
Divide by the derivative of the inner function -3.
e.g. 2.
( Avoiding “piled up” fractions )

29. e.g. 3.
This is related to
So,