1. Integration L.O. All pupils understand what integration is
All pupils can integrate individual terms

2. Recap of differentiation
If I wanted to differentiate the following what should I do?
𝑦= 𝑥 2
3) 𝑦= 2 𝑥 5 +3 𝑥 7 𝑥 3
1) 𝑦= 1 2 𝑥 2 +5𝑥
2) 𝑦= 1 𝑥 3
4) 𝑦= 𝑥 2 +3

3. Integration L.O. All pupils understand what integration is
All pupils can integrate individual terms

4. This is known as integrating!
Main 1:
what integration is
If we wanted to differentiate we would do the following: xn xn
multiply by the power
reduce the power by 1
nxn-1
Integration is the opposite of differentiation.
divide by the power
increase the power by 1 xn
This is known as integrating!

5. This is known as integrating!
Main 1:
what integration is
Notation:
𝑥 2 𝑑𝑥
This is the symbol we use to show that we are integrating.
It’s like a stretched out S
We are integrating the function x2 with respect to x
divide by the power
increase the power by 1 xn
This is known as integrating!

6. Main 1: We need to integrate to find y! 𝑑𝑦 𝑑𝑥 = 𝑥 2
what integration is
Example 1
𝑑𝑦 𝑑𝑥 = 𝑥 2
Integration is the opposite of differentiation, so it allows us to find what y was originally equal to.
We need to integrate to find y!

7. Main 1:
what integration is
Example 1
𝑥 2 𝑑𝑥
Lets check!
𝑑𝑦 𝑑𝑥 = 𝑥 2

8. Main 1: 𝑦= 𝑥 3 3 +2 what integration is
However our original equation could have been
𝑦= 𝑥
If we differentiate we get
But if we integrate that we get
Which isn’t our original equation!

9. 𝑥 2 𝑑𝑥 = 𝑥 3 3 +𝑐 Main 1: what integration is
So we have to write a general solution,
𝑥 2 𝑑𝑥 = 𝑥 𝑐
This ‘plus c’ generalises our integration, next lesson we will find c!

10. Main 1:
what integration is
Example 2
3𝑥 𝑑𝑥

11. Main 1:
what integration is
Example 2
8 𝑥 𝑥 𝑑𝑥

12. Main 1:
what integration is
Have a go in pairs
6 𝑥 𝑥 3 −4𝑥 𝑑𝑥

13. Main 1:
what integration is
Example 3
𝑡 2 + 𝑡 𝑑𝑡

14. Integration L.O. All pupils understand what integration is
All pupils can integrate individual terms

15. integrate individual terms
Main 2:
integrate individual terms
Find these integrals
5) 𝑥 −4 𝑑𝑥
1) 3 𝑥 2 𝑑𝑥
6) 3 𝑥 𝑥 𝑑𝑥
2) 3𝑥 𝑑𝑥
3) 5 𝑥 4 +2 𝑑𝑥
7) ( 𝑥 2 −2) 2 𝑑𝑥
8) 𝑥 𝑑𝑥
4) 𝑥 −2 𝑑𝑥

16. integrate individual terms
Main 2:
integrate individual terms
Answers
5)− 𝑥 −3 3
1) 𝑥 3
2) 3 2 𝑥 2
6) 𝑥 𝑥 3 2
7) 𝑥 5 5 − 4𝑥 𝑥
3) 𝑥 5 +2𝑥
4)− 𝑥 −1
8) 3 5 𝑥 5 3

17. integrate individual terms
Main 2:
integrate individual terms
Example 4
Find y in terms of x if
𝑑𝑦 𝑑𝑥 = 4 𝑥 6 −𝑥 𝑥 3

18. integrate individual terms
Main 2:
integrate individual terms
Example 5
Find 𝑦 𝑑𝑥 given that
𝑦= 𝑥 2 +2𝑥 𝑥

19. Integration L.O. All pupils understand what integration is
All pupils can integrate individual terms