1. Methods in calculus

2. FM Methods in Calculus: inverse trig functions
KUS objectives
BAT integrate using partial fractions
Starter: express in partial fractions
1 𝑥 − 1 𝑥+1
1 𝑥 𝑥+1
2𝑥−1 𝑥 2 −4
5 4(𝑥+2) + 3 4(𝑥−2)
3𝑥−2 2 𝑥 2 +3𝑥+1
5 𝑥+1 − 7 2𝑥+1

3. Don’t mix up 1 𝑎 2 − 𝑥 2 𝑑𝑥 and 1 𝑎 2 + 𝑥 2 𝑑𝑥
WB E1 a) show that 𝑎 2 − 𝑥 2 𝑑𝑥= 1 2𝑎 ln 𝑎+𝑥 𝑎−𝑥 +𝐶 where a is a constant
1 𝑎 2 − 𝑥 2 𝑑𝑥= 1 (𝑎+𝑥)(𝑎−𝑥) 𝑑𝑥
1 (𝑎+𝑥)(𝑎−𝑥) = 1 2𝑎 1 (𝑎+𝑥) + 1 (𝑎−𝑥)
= 1 2a (𝑎+𝑥) 𝑑𝑥 + 1 2𝑎 (𝑎−𝑥) 𝑑𝑥
= 1 2𝑎 ln 𝑎+𝑥 + ln 𝑎−𝑥 +𝐶
= 1 2𝑎 ln 𝑎+𝑥 𝑎−𝑥 +𝐶
Don’t mix up 𝑎 2 − 𝑥 2 𝑑𝑥 and 𝑎 2 + 𝑥 2 𝑑𝑥

4. 5 𝑥 2 +𝑥−10 (𝑥+1) 𝑥 2 −3 = 𝑨 𝑥+1 + 𝑩𝑥+𝐶 𝑥 2 −3
Notes partial fractions and quadratic factors
When a partial fraction includes a quadratic factor of the form 𝑥 2 +𝑐
we need to put a linear numerator of the form 𝐴𝑥+𝐵 on the partial fraction
5 𝑥 2 +𝑥−10 (𝑥+1) 𝑥 2 −3 = 𝑨 𝑥+1 + 𝑩𝑥+𝐶 𝑥 2 −3
WB E2
5 𝑥 2 +𝑥−10=𝑨 𝑥 2 −3 + 𝑩𝑥+𝑪 𝑥+1
Let x=−1 then
5+(−1)−10=𝑨 1−3 gives 𝑨=𝟑
Look at coefficients of 𝑥 2
5 𝑥 2 =𝑨 𝑥 2 +𝑩 𝑥 gives 𝑩=𝟐
𝑥=𝑩𝑥+𝑪𝑥 gives 𝑪=−𝟏
Look at coefficients of 𝑥
5 𝑥 2 +𝑥−10 (𝑥+1) 𝑥 2 −3 = 𝟑 𝑥+1 + 𝟐𝑥−𝟏 𝑥 2 −3

5. = 1 18 ln 𝑥 2 𝑥 2 +9 − 1 3 arctan 𝑥 3 +𝐶 So 𝐴= 1 18 𝐵= 1 3
WB E3 Show that 𝑥 𝑥 3 +9𝑥 𝑑𝑥=𝐴 ln 𝑥 2 𝑥 𝐵𝑎𝑟𝑐𝑡𝑎𝑛 𝑥 3 +𝐶 where A and B are constants to be found
1+𝑥 𝑥 3 +9𝑥 𝑑𝑥= 1+𝑥 𝑥(𝑥 2 +9) 𝑑𝑥
1+𝑥 𝑥(𝑥 2 +9) = 𝑥 + 𝑥−9 𝑥 2 +9
= 𝑥 𝑑𝑥 𝑥 𝑥 2 +9 𝑑𝑥 − 𝑥 2 +9 𝑑𝑥
1 𝑎 2 + 𝑥 2 𝑑𝑥= 1 𝑎 arctan 𝑥 𝑎 +𝐶
= 1 9 ln 𝑥 ln 𝑥 arctan 𝑥 3 +C
= ln 𝑥 ln 𝑥 arctan 𝑥 3 +𝐶
= ln 𝑥 2 𝑥 − 1 3 arctan 𝑥 3 +𝐶 So 𝐴= 𝐵= 1 3

6. Look at coefficients of 𝑥 4
WB E4 a) express 𝑥 4 +𝑥 𝑥 4 +5 𝑥 as partial fractions
b) Hence find 𝑥 4 +𝑥 𝑥 4 +5 𝑥 2 +6 𝑑𝑥
𝑥 4 +𝑥 𝑥 4 +5 𝑥 2 +6 = 𝑥 4 +𝑥 𝑥 𝑥 =A+ 𝐵𝑥+𝐶 𝑥 𝐷𝑥+𝐸 𝑥 2 +3
𝑥 4 +𝑥=𝐴 𝑥 𝑥 𝐵𝑥+𝐶 𝑥 (𝐷𝑥+𝐸) 𝑥 2 +2
Look at coefficients of 𝑥 4
𝑥 4 =𝑨 𝑥 gives 𝑨=𝟏
Look at coefficients of 𝑥 3
0=𝐵 𝑥 3 +𝐷 𝑥 gives 𝑩+𝑫=𝟎
Look at coefficients of 𝑥
𝑥=3𝐵𝑥+2𝐷𝑥 gives 3𝑩+𝟐𝑫=𝟏
Solve these simultaneous equations gives 𝑩=𝟏 𝑫=−𝟏
Look at coefficients of 𝑥 2
0=5𝐴 𝑥 2 +𝐶 𝑥 2 +𝐸 𝑥 2 gives C+𝐸=−𝟓
Look at constant terms
0=6𝐴+3𝐶+2𝐸 gives 3C+2𝐸=−𝟔
Solve these simultaneous equations gives 𝑪=𝟒 𝑬=−𝟗

7. WB E5 a) express 𝑥 4 +𝑥 𝑥 4 +5 𝑥 2 +6 as partial fractions
b) Hence find 𝑥 4 +𝑥 𝑥 4 +5 𝑥 2 +6 𝑑𝑥
𝑥 4 +𝑥 𝑥 4 +5 𝑥 2 +6 = 𝑥 4 +𝑥 𝑥 𝑥 =1+ 𝑥+4 𝑥 −𝑥−9 𝑥 2 +3
𝑏) 𝑥 4 +𝑥 𝑥 4 +5 𝑥 2 +6 𝑑𝑥 = 𝑑𝑥+ 𝑥 𝑥 𝑑𝑥 𝑥 𝑑𝑥− 𝑥 𝑥 𝑑𝑥 − 𝑥 𝑑𝑥
= 𝑥 ln 𝑥 arctan 𝑥 − 1 2 ln 𝑥 − arctan 𝑥 𝐶
= 𝑥 ln 𝑥 𝑥 arctan 𝑥 − arctan 𝑥 𝐶
NOW DO EX 3E

8. One thing to improve is –
KUS objectives
BAT integrate using partial fractions
self-assess
One thing learned is –
One thing to improve is –

9. END