1. Integration
Trig identities

2. sin 2𝑥 = 𝑠𝑒𝑐 2 𝑥 = 𝑐𝑜𝑠 2 𝑥=𝑎+𝑏 cos 2𝑥 Integration I
KUS objectives
BAT Use standard Trig identities to integrate functions
Starter: copy and complete the identities
sin 2𝑥 =
𝑠𝑒𝑐 2 𝑥 =
𝑐𝑜𝑠 2 𝑥=𝑎+𝑏 cos 2𝑥

3. WB 1a integrate standard functions You met the following in differentiation:

4. Therefore, you already can deduce the following
WB 1b
Therefore, you already can deduce the following
The modulus sign is used here to avoid potential problems with negative numbers…
(More info on the next slide!)

5. 𝑓 ′ 𝑥 𝑓 𝑥 𝑑𝑥= ln 𝑓(𝑥) +𝐶 So far we have these ‘rules’
𝑓′(𝑎𝑥+𝑏) 𝑑𝑥= 1 𝑎 𝑎𝑥+𝑏 +𝐶
1) Integrate the function using what you know from differentiation
2) Divide by the coefficient of x
𝑓 ′ 𝑥 𝑓 𝑥 𝑑𝑥= ln 𝑓(𝑥) +𝐶
3) Simplify if possible and add C
These apply ALL the functions f(x) you know

6. WB 2a Find the following integrals a) cos (2𝑥+3) 𝑑𝑥
WB 2a Find the following integrals a) cos (2𝑥+3) 𝑑𝑥 b) 𝑠𝑒𝑐 2 3𝑥 𝑑𝑥 c) cos 4𝑥 + 3 𝑥 𝑑𝑥 d) sec 2𝑥 tan 2𝑥 𝑑𝑥
Consider starting with sin(2x + 3) and the answer that would give
𝑑 𝑑𝑥 ( sin (2𝑥+3))=2 cos⁡ (2𝑥+3)
This is double what we are wanting to integrate
 Therefore, we must ‘start’ with half the amount…
cos (2𝑥+3) 𝑑𝑥 = 1 2 sin (2𝑥+3) + 𝐶
This is a VERY common method of integration – considering what we might start with that would differentiate to our answer…

7. WB 2b Find the following integrals a) cos (2𝑥+3) 𝑑𝑥
WB 2b Find the following integrals a) cos (2𝑥+3) 𝑑𝑥 b) 𝑠𝑒𝑐 2 3𝑥 𝑑𝑥 c) cos 4𝑥 + 3 𝑥 𝑑𝑥 d) sec 2𝑥 tan 2𝑥 𝑑𝑥
𝑑 𝑑𝑥 ( tan 3𝑥)=3 𝑠𝑒𝑐 2 3𝑥
Consider starting with tan3x and the answer that would give
This is three times what we are wanting to integrate
 Therefore, we must ‘start’ with a third of the amount…
𝑠𝑒𝑐 2 3𝑥 𝑑𝑥 = 1 3 tan 3𝑥 + 𝐶

8. WB 2c Find the following integrals a) cos (2𝑥+3) 𝑑𝑥
WB 2c Find the following integrals a) cos (2𝑥+3) 𝑑𝑥 b) 𝑠𝑒𝑐 2 3𝑥 𝑑𝑥 c) cos 4𝑥 + 3 𝑥 𝑑𝑥 d) sec 2𝑥 tan 2𝑥 𝑑𝑥
𝑑 𝑑𝑥 ( sin 4𝑥)=4 cos 4𝑥
𝑑 𝑑𝑥 ( ln 𝑥)= 1 𝑥
2 cos 4𝑥 + 3 𝑥 𝑑𝑥 =− 1 2 sin 4𝑥 +3 ln 𝑥 +𝐶

9. WB 2d Find the following integrals a) cos (2𝑥+3) 𝑑𝑥
WB 2d Find the following integrals a) cos (2𝑥+3) 𝑑𝑥 b) 𝑠𝑒𝑐 2 3𝑥 𝑑𝑥 c) cos 4𝑥 + 3 𝑥 𝑑𝑥 d) sec 2𝑥 tan 2𝑥 𝑑𝑥
𝑑 𝑑𝑥 ( sec 𝑥)= sec 𝑥 tan 𝑥
sec 2𝑥 tan 2𝑥 𝑑𝑥 = ½ sec 2𝑥 + 𝐶

10. Write as many Trig Identities as you can from memory
𝐭𝐚𝐧 𝒙 = 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝐬𝐞𝐜 𝒙 = 𝟏 𝐜𝐨𝐬 𝒙
𝒔𝒊𝒏 𝟐 𝒙+ 𝒄𝒐𝒔 𝟐 𝒙=𝟏
𝐜𝐨𝐬𝐞𝐜 𝒙 = 𝟏 𝐬𝐢𝐧 𝒙
𝐜𝐨𝐭 𝒙 = 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒙
𝒕𝒂𝒏 𝟐 𝒙+𝟏= 𝒔𝒆𝒄 𝟐 𝒙
𝟏+ 𝒄𝒐𝒕 𝟐 𝒙= 𝒄𝒐𝒔𝒆𝒄 𝟐 𝒙
sin (𝐴±𝐵) =sin A cos B±sin B cos A
cos (𝐴±𝐵) =cos A cos B∓sin A sin B
𝐜𝐨𝐬 𝟐𝒙 = 𝒄𝒐𝒔 𝟐 𝒙− 𝒔𝒊𝒏 𝟐 𝒙
tan (𝐴±𝐵) = tan 𝐴 ± tan 𝐵 1∓ tan 𝐴 tan 𝐵
𝐜𝐨𝐬 𝟐𝒙 =𝟏− 𝟐 𝒔𝒊𝒏 𝟐 𝒙
𝐜𝐨𝐬 𝟐𝒙 = 𝟐𝒄𝒐𝒔 𝟐 𝒙−𝟏
sin 𝐴 + sin 𝐵 =2sin 𝐴+𝐵 2 cos 𝐴−𝐵 2
𝐬𝐢𝐧 𝟐𝒙 =𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
sin 𝐴 − sin 𝐵 =2cos 𝐴+𝐵 2 sin 𝐴−𝐵 2
𝟏 𝟐 + 𝟏 𝟐 𝐜𝐨𝐬 𝟐𝒙 = 𝒄𝒐𝒔 𝟐 𝒙
cos 𝐴 + cos 𝐵 =2 𝑐𝑜𝑠 𝐴+𝐵 2 cos 𝐴−𝐵 2
𝟏 𝟐 − 𝟏 𝟐 𝐜𝐨𝐬 𝟐𝒙 = 𝒔𝒊𝒏 𝟐 𝒙
cos 𝐴 − cos 𝐵 =−2sin 𝐴+𝐵 2 sin 𝐴−𝐵 2

11. cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥 = 1 sin 𝑥 × cos 𝑥 sin 𝑥 − 2 𝑒 𝑥
WB 3a Find the following integrals a) cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥 b) 𝑡𝑎𝑛 2 𝑥 𝑑𝑥 c) 𝑠𝑖𝑛 2 𝑥 𝑑𝑥 d) sin 3𝑥 cos 3𝑥 𝑑𝑥
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
𝑑 𝑑𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥 cot 𝑥 =−𝑐𝑜𝑠𝑒𝑐 𝑥
cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥 = sin 𝑥 × cos 𝑥 sin 𝑥 − 2 𝑒 𝑥
= 𝑐𝑜𝑠𝑒𝑐 𝑥 cot 𝑥 − 2 𝑒 𝑥
=−𝑐𝑜𝑠𝑒𝑐 𝑥 −2 𝑒 𝑥 +𝐶

12. WB 3b Find the following integrals a) cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥
WB 3b Find the following integrals a) cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥 b) 𝑡𝑎𝑛 2 𝑥 𝑑𝑥 c) 𝑠𝑖𝑛 2 𝑥 𝑑𝑥 d) sin 3𝑥 cos 3𝑥 𝑑𝑥
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
𝑑 𝑑𝑥 𝑠𝑒𝑐 2 𝑥 =𝑡𝑎𝑛 𝑥
𝑡𝑎𝑛 2 𝑥 𝑑𝑥 = 𝑠𝑒𝑐 2 𝑥 −1 𝑑𝑥
= tan 𝑥 −𝑥 +𝐶

13. WB 3c Find the following integrals a) cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥
WB 3c Find the following integrals a) cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥 b) 𝑡𝑎𝑛 2 𝑥 𝑑𝑥 c) 𝑠𝑖𝑛 2 𝑥 𝑑𝑥 d) sin 3𝑥 cos 3𝑥 𝑑𝑥
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
𝑑 𝑑𝑥 ( sin 2𝑥)=2 cos⁡ 2𝑥
𝑠𝑖𝑛 2 𝑥 𝑑𝑥 = − 1 2 cos 2𝑥 𝑑𝑥
= 1 2 𝑥 − 1 4 sin 2𝑥 +𝐶

14. WB 3d Find the following integrals a) cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥
WB 3d Find the following integrals a) cos 𝑥 𝑠𝑖𝑛 2 𝑥 −2 𝑒 𝑥 𝑑𝑥 b) 𝑡𝑎𝑛 2 𝑥 𝑑𝑥 c) 𝑠𝑖𝑛 2 𝑥 𝑑𝑥 d) sin 3𝑥 cos 3𝑥 𝑑𝑥
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
𝑑 𝑑𝑥 ( cos 6𝑥)=−6 sin⁡6𝑥
sin 3𝑥 𝑐𝑜𝑠3𝑥 𝑑𝑥 = sin 6𝑥 𝑑𝑥
=− cos 6𝑥 +𝐶

15. Find the following integral:
WB 4a
Find the following integral:
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
Expand the bracket
Replace tan2x
Simplify
Integrate separately

16. WB 4b Integrate the following:
WB 4b Integrate the following:
In this case consider the power of sine.
 If it has been differentiated, it must have been sin3x originally…
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
Write as a cubed bracket
Differentiate using the chain rule
Rewrite – this has given us exactly what we wanted!
Don’t forget the + C!

17. WB 4c Integrate the following:
WB 4c Integrate the following:
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
Write using powers
Imagine how we could end up with a -3 as a power...
Use the chain rule
Rewrite
This is double what we want so multiply the ‘guess’ by 1/2

18. WB 4d Integrate the following:
WB 4d Integrate the following:
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
Consider using a power 5
Write as a bracket to the power 5
Differentiate using the chain rule
We have an extra secx
HOWEVER:
We cannot just add this to our ‘guess’ as before, as the differentiation will need to be performed using the product rule , rather than the Chain rule!
We need to find another way!

19. WB 4d (cont) Integrate the following:
Consider using a power 4
Write as a bracket to the power 4
Differentiate using the chain rule
This is what we want, but 4/5 of the amount
 Multiply by the guess by 5/4

20. We can also use the formulas from the formula booklet
𝐬𝐢𝐧 (𝑨±𝑩) =𝐬𝐢𝐧 𝐀 𝐜𝐨𝐬 𝐁±𝐬𝐢𝐧 𝐁 𝐜𝐨𝐬 𝐀
𝐜𝐨𝐬 (𝑨±𝑩) =𝐜𝐨𝐬 𝐀 𝐜𝐨𝐬 𝐁∓𝐬𝐢𝐧 𝐀 𝐬𝐢𝐧 𝐁
𝐭𝐚𝐧 (𝑨±𝑩) = 𝐭𝐚𝐧 𝑨 ± 𝐭𝐚𝐧 𝑩 𝟏∓ 𝐭𝐚𝐧 𝑨 𝐭𝐚𝐧 𝑩
𝐬𝐢𝐧 𝑨 + 𝐬𝐢𝐧 𝑩 =𝟐𝐬𝐢𝐧 𝑨+𝑩 𝟐 𝐜𝐨𝐬 𝑨−𝑩 𝟐
𝐬𝐢𝐧 𝑨 − 𝐬𝐢𝐧 𝑩 =𝟐𝐜𝐨𝐬 𝑨+𝑩 𝟐 𝐬𝐢𝐧 𝑨−𝑩 𝟐
𝐜𝐨𝐬 𝑨 + 𝐜𝐨𝐬 𝑩 =𝟐 𝒄𝒐𝒔 𝑨+𝑩 𝟐 𝐜𝐨𝐬 𝑨−𝑩 𝟐
𝐜𝐨𝐬 𝑨 − 𝐜𝐨𝐬 𝑩 =−𝟐𝐬𝐢𝐧 𝑨+𝑩 𝟐 𝐬𝐢𝐧 𝑨−𝑩 𝟐
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥

21. show that 2sin 3𝑥cos 𝑥 = sin 4𝑥 + sin 2𝑥 b) Find sin 3𝑥 cos 𝑥 𝑑𝑥
WB 5
show that 2sin 3𝑥cos 𝑥 = sin 4𝑥 + sin 2𝑥
b) Find sin 3𝑥 cos 𝑥 𝑑𝑥
sin 𝐴 + sin 𝐵 =2 sin 𝐴+𝐵 2 cos 𝐴−𝐵 2
𝐴+𝐵 2 =3𝑥
𝐴+𝐵=6𝑥
𝐴=4𝑥 𝐵=2𝑥
𝐴−𝐵 2 =𝑥
𝐴−𝐵=2𝑥
2sin 3𝑥 cos 𝑥 = sin 4𝑥 + sin 2𝑥 𝑄𝐸𝐷
sin 3𝑥 cos 𝑥 𝑑𝑥 = sin 4𝑥 𝑑𝑥 sin 2𝑥 𝑑𝑥
=− 1 8 cos 4𝑥 − 1 4 cos 2𝑥 +𝐶

22. WB 6 Find cos 𝑥 −2 2 𝑑𝑥 1 2 cos 2𝑥 −4 cos 𝑥 + 9 2 𝑑𝑥
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
cos 𝑥 −2 2 = 𝑐𝑜𝑠 2 𝑥 −4 cos 𝑥 +4
cos 𝑥 −2 2 = cos 2𝑥 −4 cos 𝑥 +4
cos 𝑥 −2 2 = 1 2 cos 2𝑥 −4 cos 𝑥
1 2 cos 2𝑥 −4 cos 𝑥 𝑑𝑥
= 1 4 𝑠𝑖𝑛 2𝑥 −4 sin 𝑥 𝑥 +𝐶

23. WB 7 Find 𝑠𝑖𝑛 4 𝑥 𝑑𝑥 3 8 − cos 2𝑥 + 1 8 cos 4𝑥 𝑑𝑥
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
= − 1 2 cos 2𝑥 2
𝑠𝑖𝑛 4 𝑥 = 𝑠𝑖𝑛 2 𝑥 2
𝑠𝑖𝑛 4 𝑥= 1 4 − cos 2𝑥 𝑐𝑜𝑠 2 2𝑥
𝑠𝑖𝑛 4 𝑥= 1 4 − cos 2𝑥 cos 4𝑥
3 8 − cos 2𝑥 cos 4𝑥 𝑑𝑥
= 3 8 𝑥− 1 2 𝑠𝑖𝑛 2𝑥 sin 4𝑥 +𝐶

24. WB 8 Find 𝑐𝑜𝑠 2 5 2 − 𝑥 2 𝑑𝑥 1 2 − 1 2 cos 5−𝑥 𝑑𝑥
𝑠𝑖𝑛 2 𝑥+ 𝑐𝑜𝑠 2 𝑥=1
𝑡𝑎𝑛 2 𝑥+1= 𝑠𝑒𝑐 2 𝑥
1+ 𝑐𝑜𝑡 2 𝑥= 𝑐𝑜𝑠𝑒𝑐 2 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛 2 𝑥
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 = 𝑐𝑜𝑠 2 𝑥
1 2 − 1 2 cos 2𝑥 = 𝑠𝑖𝑛 2 𝑥
𝑐𝑜𝑠 − 1 2 𝑥 = 1 2 − 1 2 cos 5−𝑥
1 2 − 1 2 cos 5−𝑥 𝑑𝑥
= 1 2 𝑥 𝑠𝑖𝑛 5−𝑥 𝐶

25. The formula booklet gives you some help
We need more techniques before
meeting all of these

26. One thing to improve is –
KUS objectives BAT Use standard Trig identities to integrate functions
self-assess
One thing learned is –
One thing to improve is –

27. END