1. “Teach A Level Maths” Vol. 1: AS Core Modules
50: Harder Indefinite Integration
© Christine Crisp

2. Module C2
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3. n does not need to be an integer BUT notice that the rule is for
Reminder:
add 1 to the power
divide by the new power
add C
n does not need to be an integer BUT notice that the rule is for
It cannot be used directly for terms such as

4. e.g.1 Evaluate
Solution: Using the law of indices,
So,
This minus sign . . .
. . . makes the term negative.

5. e.g.1 Evaluate
Solution: Using the law of indices,
So,
But this one . . .
is an index

6. e.g.2 Evaluate
Solution:
We need to simplify this “piled up” fraction.
Multiplying the numerator and denominator by 2 gives
We can get this answer directly by noticing that . . .
. . . dividing by a fraction is the same as multiplying by its reciprocal. ( We “flip” the fraction over ).

7. e.g.2 Evaluate
Solution:
We need to simplify this “piled up” fraction.
Multiplying the numerator and denominator by 2 gives
We can get this answer directly by noticing that . . .
. . . dividing by a fraction is the same as multiplying by its reciprocal. ( We “flip” the fraction over ).

8. e.g.3 Evaluate
Solution:
So,
Using the law of indices,

9. e.g.4 Evaluate
Solution:
We cannot integrate with x in the denominator.
Write in index form
Split up the fraction
Use the 2nd law of indices:

10. e.g.4 Evaluate
Solution:
and
The terms are now in the form where we can use our rule of integration.
Instead of dividing by ,multiply by
Instead of dividing by ,multiply by

11. ( 1, 0 ) on the curve: e.g.5 The curve passes through the point
( 1, 0 ) and
Find the equation of the curve.
Solution:
It’s important to prepare all the terms before integrating any of them
( 1, 0 ) on the curve:
So the curve is

12. Exercise
Evaluate 1. 2.
Solution:

13. ( 2, 0 ) on the curve: Exercise
3. Given that , find the equation of the curve through the point ( 2, 0 ).
Solution:
( 2, 0 ) on the curve:
So the curve is

15. 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.

16. e.g.1 Evaluate
Solution: Using the law of indices,
So,
This minus sign . . .
. . . makes the term negative.
But this one is an index

17. e.g.2 Evaluate
We need to simplify this “piled up” fraction.
Multiplying the numerator and denominator by 2 gives
We can get this answer directly by noticing that . . .
. . . dividing by a fraction is the same as multiplying by its reciprocal. ( We “flip” the fraction over ).
Solution:

18. e.g.3 Evaluate
Solution:
So,
Using the law of indices,

19. e.g.4 Evaluate
Solution:
Write in index form
Split up the fraction
We cannot integrate with x in the denominator.
Use the laws of indices: and

20. The terms are now in the form where we can use our rule of integration.

21. ( 1, 0 ) on the curve: e.g.5 The curve passes through the point
Solution:
e.g.5 The curve passes through the point
( 1, 0 ) and
Find the equation of the curve.
( 1, 0 ) on the curve:
So the curve is
It’s important to prepare all the terms before integrating any of them