1. Integration

2. Integration I Starter: KUS objectives
BAT Integrate using a substitution
BAT find a definite integral using a substitution, including converting the bounds
Starter:

3. WB 9a Find tan 𝑥 𝑑𝑥 using the substitution 𝑢= cos 𝑥
To integrate this, you need to replace the x terms with equivalent u terms, and replace the dx with an equivalent du
= sin 𝑥 cos 𝑥 𝑑𝑥
𝒖=𝒄𝒐𝒔 𝒙
𝒙=𝐮 −𝟏
=− 1 𝑢 𝑑𝑢
𝒅𝒖 𝒅𝒙 =−𝒔𝒊𝒏 𝒙
𝒅𝒖=−𝒔𝒊𝒏 𝒙 𝒅𝒙
=− ln cos 𝑥 +𝐶
= ln ( cos 𝑥) −1 +𝐶 = ln sec 𝑥 +𝐶

4. WB 9b Find cot 𝑥 𝑑𝑥 = cos 𝑥 sin 𝑥 𝑑𝑥 𝒖= 𝒔𝒊𝒏 𝒙 = 1 𝑢 𝑑𝑢 𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙
= 𝑢 𝑑𝑢
𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙
= ln 𝑢 +𝐶
𝒅𝒖=𝒄𝒐𝒔 𝒙 𝒅𝒙
= ln sin 𝑥 +𝐶

5. WB 10 Find cos 𝑥 sin 𝑥 1+ sin 𝑥 3 𝑑𝑥
using the substitution 𝑢= sin 𝑥 +1
sin 𝑥 1+ sin 𝑥 3 cos 𝑥 𝑑𝑥
𝒖= 𝒔𝒊𝒏 𝒙+𝟏
= 𝑢−1 𝑢 3 𝑑𝑢
𝒖−𝟏=𝒔𝒊𝒏 𝒙
= 𝑢 4 − 𝑢 3 𝑑𝑢
𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙
= 1 5 𝑢 5 − 1 4 𝑢 4 +𝐶
𝒅𝒖=𝒄𝒐𝒔 𝒙 𝒅𝒙
= 𝒔𝒊𝒏 𝒙+𝟏 5 − 𝒔𝒊𝒏 𝒙+𝟏 4 +𝐶

6. WB 11 Show that 0 𝜋/2 cos 𝑥 3+2 sin 𝑥 𝑑𝑥 = 1 2 ln 5 3
= sin 𝑥 2 cos 𝑥 𝑑𝑥
𝒖= 𝟑+𝟐 𝒔𝒊𝒏 𝒙
= 𝑢 𝑑𝑢
𝒅𝒖 𝒅𝒙 =𝟐 𝒄𝒐𝒔 𝒙
= ln 𝑢 +𝐶
𝒅𝒖=𝟐 𝒄𝒐𝒔 𝒙 𝒅𝒙
= ln 3+2 sin 𝑥
= ln (3+2) − ln (3+0
= ln 5 3

7. WB 12 Find cos 𝑥 𝑠𝑖𝑛 5 𝑥 𝑑𝑥 = 𝑠𝑖𝑛 5 𝑥 cos 𝑥 𝑑𝑥 𝒖= 𝒔𝒊𝒏 𝒙 = 𝑢 5 𝑑𝑢
𝒖= 𝒔𝒊𝒏 𝒙
= 𝑢 5 𝑑𝑢
𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙
= 1 6 𝑢 6 +𝐶
𝒅𝒖= 𝒄𝒐𝒔 𝒙 𝒅𝒙
= 𝑠𝑖𝑛 6 𝑥 +𝐶

8. WB 13 Find sin 𝑥 𝑐𝑜𝑠 𝑛 𝑥 𝑑𝑥 = 𝑐𝑜𝑠 𝑛 𝑥 sin 𝑥 𝑑𝑥 𝒖= 𝒄𝒐𝒔 𝒙 =− 𝑢 𝑛 𝑑𝑢
𝒖= 𝒄𝒐𝒔 𝒙
=− 𝑢 𝑛 𝑑𝑢
𝒅𝒖 𝒅𝒙 =− 𝒔𝒊𝒏 𝒙
=− 1 𝑛+1 𝑢 6 +𝐶
𝒅𝒖=− 𝒔𝒊𝒏 𝒙 𝒅𝒙
= 1 𝑛+1 𝑐𝑜𝑠 𝑛+1 𝑥 +𝐶

9. WB 14 Find 𝑐𝑜𝑠𝑒𝑐 𝑥 𝑑𝑥 = 𝑐𝑜𝑠𝑒𝑐 2 𝑥+𝑐𝑜𝑠𝑒𝑐 𝑥 cot 𝑥 cosec 𝑥 + cot 𝑥 𝑑𝑥
Multiply through by 𝑐𝑜𝑠𝑒𝑐 𝑥+ cot 𝑥 cosec 𝑥 + cot 𝑥
=− 1 𝑢 𝑑𝑢
𝒖= 𝒄𝒐𝒔𝒆𝒄 𝒙+𝒄𝒐𝒕 𝒙
=− ln 𝑢 +𝐶
𝒅𝒖 𝒅𝒙 =−𝒄𝒐𝒔𝒆𝒄 𝒙 𝒄𝒐𝒕 𝒙 − 𝒄𝒐𝒔𝒆𝒄 𝟐 𝒙
=− ln 𝒄𝒐𝒔𝒆𝒄 𝒙+𝒄𝒐𝒕 𝒙 +𝐶
𝒅𝒖=−𝒄𝒐𝒔𝒆𝒄 𝒙 𝒄𝒐𝒕 𝒙 − 𝒄𝒐𝒔𝒆𝒄 𝟐 𝒙 dx

10. WB 15 Find 𝑠𝑒𝑐 𝑥 𝑑𝑥 = 𝑠𝑒𝑐 2 𝑥+𝑠𝑒𝑐 𝑥 tan 𝑥 sec 𝑥 + tan 𝑥 𝑑𝑥
Multiply through by 𝑠𝑒𝑐 𝑥+ tan 𝑥 sec 𝑥 + tan 𝑥
= 1 𝑢 𝑑𝑢
𝒖= 𝒔𝒆𝒄 𝒙+𝒕𝒂𝒏 𝒙
= ln 𝑢 +𝐶
𝒅𝒖 𝒅𝒙 =𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙+ 𝒔𝒆𝒄 𝟐 𝒙
= ln 𝒔𝒆𝒄 𝒙+𝒕𝒂𝒏 𝒙 +𝐶
𝒅𝒖=𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙+ 𝒔𝒆𝒄 𝟐 𝒙 dx

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

12. KUS objectives Bat Integrate using a substitution
BAT find a definite integral using a substitution, including converting the bounds
self-assess
One thing learned is –
One thing to improve is –

13. END