1. Integration

2. A2 Integration II Starter: KUS objectives
BAT use integration by parts with trig functions including ‘cyclic’ problems
Starter:

3. WB 7a find the following integrals a) 𝑥 cos 𝑥 𝑑𝑥 b) 2𝑥 sin 3𝑥 𝑑𝑥
𝑢=𝑥
𝑑𝑣 𝑑𝑥 = cos 𝑥
𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥
𝑣= sin 𝑥
𝑥 cos 𝑥 𝑑𝑥
𝑑𝑢 𝑑𝑥 =1
=𝑥 sin 𝑥 − sin 𝑥 1 𝑑𝑥
=𝑥 sin 𝑥 + cos 𝑥 +𝐶

4. WB 7b find the following integrals a) 𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙 b) 𝟐𝒙 𝐬𝐢𝐧 𝟑𝒙 𝒅𝒙
𝑢=2𝑥
𝑑𝑣 𝑑𝑥 = 𝑠𝑖𝑛 3𝑥
𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥
𝑣=− 1 3 cos 3𝑥
= 2 3 𝑥 cos 𝑥 − − 2 3 cos 3𝑥 𝑑𝑥
𝑑𝑢 𝑑𝑥 =2
= 2 3 𝑥 cos 𝑥 sin 3𝑥 +𝐶

5. WB 8 Show that 0 𝜋/2 4𝑥 sin 4𝑥 𝑑𝑥 =− 1 8 𝜋
𝑢=4𝑥
𝑣=− 1 4 cos 4𝑥
𝑑𝑢 𝑑𝑥 =4
𝑑𝑣 𝑑𝑥 = sin 4𝑥
𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥
=− 𝑥 cos 4𝑥 − − cos 4𝑥 𝑑𝑥
= − 1 4 𝑥 cos 4𝑥 sin 4𝑥 𝜋/2 0
= − 1 4 × 𝜋 − 0+0 =− 1 8 𝜋 QED

6. WB 9 Show that 𝜋/6 𝜋/3 3𝑥 cos 3𝑥 𝑑𝑥 =− 1 6 (2+𝜋)
𝑢=3𝑥
𝑣= 1 3 sin 3𝑥
𝑑𝑢 𝑑𝑥 =3
𝑑𝑣 𝑑𝑥 = cos 3𝑥
𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥
= 𝑥 sin 3𝑥 − sin 3𝑥 𝑑𝑥
= 𝑥 sin 3𝑥 𝑐𝑜𝑠 3𝑥 𝜋/3 𝜋/6
= 𝜋 −1 − 𝜋
=− 1 3 − 𝜋 6 =− 1 6 (2+𝜋) QED

7. WB 10 Circular questions Find 𝑒 𝑥 cos 𝑥 𝑑𝑥
𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥
WB Circular questions Find 𝑒 𝑥 cos 𝑥 𝑑𝑥
𝑢= 𝑒 𝑥
𝑣= sin 𝑥
𝑑𝑢 𝑑𝑥 = 𝑒 𝑥
𝑑𝑣 𝑑𝑥 = cos 𝑥
Round 1
= 𝑒 𝑥 sin 𝑥 − 𝑒 𝑥 sin 𝑥 𝑑𝑥
= 𝑒 𝑥 sin 𝑥 − − 𝑒 𝑥 cos 𝑥 − − 𝑒 𝑥 cos 𝑥 𝑑𝑥
= 𝑒 𝑥 sin 𝑥 + 𝑒 𝑥 cos 𝑥 − 𝑒 𝑥 cos 𝑥 𝑑𝑥
𝑢= 𝑒 𝑥
𝑣=− cos 𝑥
𝑑𝑢 𝑑𝑥 = 𝑒 𝑥
𝑑𝑣 𝑑𝑥 = sin 𝑥
Round 2
How is this circular?
2 𝑒 𝑥 cos 𝑥 𝑑𝑥 = 𝑒 𝑥 sin 𝑥 + 𝑒 𝑥 cos 𝑥
𝑒 𝑥 cos 𝑥 𝑑𝑥 = 1 2 𝑒 𝑥 sin 𝑥 + cos 𝑥 +C

8. Skills 219
homework 219

9. One thing to improve is –
KUS objectives BAT use integration by parts with trig functions including ‘cyclic’ problems
self-assess
One thing learned is –
One thing to improve is –

10. END