1. “Teach A Level Maths” Vol. 2: A2 Core Modules
28: Integration giving Logs
© Christine Crisp

2. Module C3 Module C4 AQA Edexcel MEI/OCR OCR
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3. When we differentiated we found the general result
with its derivative “on top”
The inner function . . .

4. When we differentiated we found the general result
This result came from using the chain rule.
e.g. Differentiate with respect to x.
Solution: Let

5. Since indefinite integration is the reverse of differentiation

6. Since indefinite integration is the reverse of differentiation
The modulus signs are there because the log of a negative number is not defined.

7. e.g. 1

8. e.g. 1

9. e.g. 1

10. e.g. 1

11. e.g. 1
e.g. 2
We don’t need the mod signs since is always positive.
e.g. 3

12. Some integrals can be adjusted so that they lead to logs.
We need 2x in the numerator . . .
so we write 2x
We have multiplied by 2, so to compensate, we must divide by 2.

13. Some integrals can be adjusted so that they lead to logs.
Now the can just stay where it is, multiplying the integral.

14. e.g. 5
What do we want in the numerator?
Ans:
We now need to get rid of the minus and replace the 3.
. . .

15. e.g. 5
What do we want in the numerator?
Ans:
We now need to get rid of the minus and replace the 3.
Always check by multiplying the numerator by the constant outside the integral:

16. e.g. 5
What do we want in the numerator?
Ans:
We now need to get rid of the minus and replace the 3.

17. e.g. 6
. . .

18. e.g. 6

19. BUT we can never adjust x
e.g.
is not

20. a multiplying constant
SUMMARY
Quotients consisting of a function with its derivative “on top” can be integrated directly.
The integral is the natural log of
Quotients can be adjusted only if
a multiplying constant
needs to be changed.

21. Exercises
Integrate the following: 1. 2. 3. 4. 5. 6.
( I haven’t made a mistake here ! )

22. Solutions: 1. 2.
Mod signs here aren’t wrong but are unnecessary. 3. 4.

23. Solutions: 5. 6.
Using the 3rd log law we can write this as
giving

24. Definite integration
There is no difference in method but we are expected to give exact answers and to simplify the logs.
e.g. 1
The curved brackets are essential

25. e.g. 2
We have to be very careful with this minus sign
The curved brackets are essential

26. Exercises
Evaluate the following: 1. 2.

27. 1.
Solutions:

28. Solutions: 2.

30. 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.

31. SUMMARY
Quotients consisting of a function with its derivative “on top” can be integrated directly.
The integral is the natural log of
Quotients can be adjusted if only a multiplying constant needs to be changed.

32. e.g. 1
e.g. 2
e.g. 3

33. e.g. 4
e.g. 5
e.g. 6

34. Definite integration
There is no difference in method but we are expected to give exact answers and to simplify the logs.
e.g. 1
The curved brackets are essential

35. e.g. 2
We have to be very careful with this minus sign