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P3 Chapter 7 :: Integration

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Overview

In this chapter, you’ll be able to integrate a significantly greater variety of expressions, and be able to solve differential equations.

Integration by ‘reverse chain rule’

(We imagine what would have differentiated to get the expression.)

Integration by standard result

(There’s certain expressions you’re expected to know straight off.)

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SKILL #1: Integrating Standard Functions

There’s certain results you should be able to integrate straight off, by just thinking about the opposite of differentiation.

Remember my memorisation trick of picturing sin above cos from C3, so that ‘going down’ is differentiating and ‘going up’ is integrating, and we change the sign if the wrong way round.

It’s vital you remember this one.

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Have a good stare at this slide before turning your paper over – let’s see how many you remember…

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Quickfire Questions (without cheating!)

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Quickfire Questions (without cheating!)

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Quickfire Questions (without cheating!)

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Test Your Understanding

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Hint: What ‘reciprocal’ trig functions does this simplify to?

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Exercise 7A

Pearson Pure Mathematics 3

Pages 148-149

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Quickfire:

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Check Your Understanding

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Exercise 7B

Pages 150-151

a

c

e

f

g

h

1

a

2

b

c

d

e

a

b

c

d

f

h

j

a

c

3

4

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SKILL #3: Integrating using Trig Identities

Q

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Q

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Check Your Understanding

Q

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Q

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Exercise 7C

Pearson Pure Mathematics Year 3

Page 153

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SKILL #4: Reverse Chain Rule

There’s certain more complicated expressions which look like the result of having applied the chain rule. I call this process ‘consider then scale’:

Consider some expression that will differentiate to something similar to it.
Differentiate, and adjust for any scale difference.

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SKILL #4: Reverse Chain Rule

In words: “If the bottom of a fraction differentiates to give the top (forgetting scaling), try ln of the bottom”.

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Quickfire

In your head!

Not in your head…

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Test Your Understanding

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Exercise 7D

Pearson Pure Mathematics 3

Page 155-156

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SKILL #8: Integrating top-heavy algebraic fractions

How would we deal with this? (the clue’s in the title)

Some manipulation to simplify

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Now integrate

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Test Your Understanding

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	How to deal with it		Formula booklet?
	Standard result		No
	Standard result		No
			Yes
			No
			No
			No
			Yes
			Yes
			Yes

Summary of Functions

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Summary of Functions

	How to deal with it		Formula booklet?
	By observation.		No!
	By observation.		Yes (but memorise)
			No
			No
			No
			No

	Use partial fractions.

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	How to deal with it




	Reverse chain rule.

Summary of Functions

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