1. Integration 2a

2. BAT Find integrals using rule for log functions
Integration I
KUS objectives
BAT Find integrals using rule for log functions
BAT Solve problems where partial factions used with this rule;
Starter:
Differentiate 𝑥−3 7
Integrate 𝑥−3 7
Integrate 𝑒 4𝑥+7

3. Therefore you can use either x or –x in the Integral
Notes
As we are integrating to find the Area, you can see for any 2 points, the area will be the same for either graph…
Therefore you can use either x or –x in the Integral
However, you cannot find ln of a negative, just use the positive value instead!
This is saying when we Integrate either of the following, we get the same result:

4. WB 5 Find the following integral 1 3𝑥 + 2 𝑑𝑥
1) Integrate the function using what you know from differentiation
2) Divide by the coefficient of x
3) Simplify if possible and add C
We can extend the rule
𝑓′(𝑎𝑥+𝑏) 𝑑𝑥= 1 𝑎 𝑎𝑥+𝑏 +𝐶 To
𝑓 ′ 𝑥 𝑓 𝑥 𝑑𝑥= ln 𝑓(𝑥) +𝐶

5. WB 6 Find the following integrals: a) 1 4𝑥−5 𝑑𝑥 b) 3 3𝑥+11 𝑑𝑥 c) −2 7−6𝑥 𝑑𝑥
1 4𝑥−5 𝑑𝑥 = ln 4𝑥−5 +𝐶
3 3𝑥+11 𝑑𝑥 = ln 3𝑥 C
= ln 3𝑥+11 +𝐶
−2 7−6𝑥 𝑑𝑥 = 1 −6 −2 ln 7−6𝑥 +C
= ln 7−6𝑥 +𝐶

6. WB 7 𝑓 ′ 𝑥 = 5 5𝑥−1 , Given that the curve 𝑓(𝑥) passes through point (8, ln 16 ) find 𝑓(𝑥)
Using point (8, ln 3 ) ln 16 = ln 8 +𝐶
𝐶 = ln 16 − ln 8
= ln 2
So (8, ln 3 ) 𝑓 𝑥 = ln 5𝑥−1 + ln 2

7. WB 8 Show that 2 7 2 4𝑥−3 𝑑𝑥 = ln 5 2 7 2 4𝑥−3 𝑑𝑥 = 1 4 2 ln 4𝑥−3 7 2

8. Skills 212
homework 212

9. WB 9 Find the following integral 𝑥−5 𝑥+1 𝑥−2 𝑑𝑥
Let x = 2
Let x = -1

10. WB 10a Find the following integral 𝑥 2 −3𝑥+2 9 𝑥 2 −4 𝑑𝑥 by rearranging to the form 𝑁+ 𝐴 3𝑥+2 + 𝐵 3𝑥−2

11. Find the following integral:
WB 10b
Find the following integral:
=𝑥 ln 3𝑥−2 3𝑥 𝐶

12. WB 11a Find the following integral 3 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 𝑑𝑥 by rearranging to partial fractions
3 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 = 𝐴 𝑥−2 + 𝐵 𝑥+1 + 𝐶 (𝑥+1) 2
=𝐴( 𝑥 𝐵 𝑥−2 𝑥+1 +𝐶(𝑥−2)
𝑥=2 gives 9A= solves to 𝐴=1
𝑥=−1 gives -3C= solves to C=4
𝑥=0 gives A-2B-2C=-11 solves to B=2
3 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 = 1 𝑥−2 + 2 𝑥 (𝑥+1) 2

13. = 1 𝑥−2 + 2 𝑥+1 + 4 (𝑥+1) 2 𝑑𝑥 = ln 𝑥−2 +2 ln 𝑥+1 −4 𝑥+1 −1 +C
WB 11b Find the following integral 𝑥 2 +4𝑥−11 (𝑥−2) (𝑥+1) 2 𝑑𝑥 by rearranging to partial fractions
= 𝑥−2 + 2 𝑥 (𝑥+1) 2 𝑑𝑥
A knotty integral!
‘KNOT’ a logarithm term
= ln 𝑥−2
+2 ln 𝑥+1
−4 𝑥+1 −1 +C

14. WB 12 Show that 5 6 3𝑥+2 (𝑥+3)(𝑥−4) 𝑑𝑥= ln 9 2
3𝑥+2 (𝑥+3)(𝑥−4) = …= 1 𝑥 𝑥−4
5 6 3𝑥+2 (𝑥+3)(𝑥−4) 𝑑𝑥 = 𝑥 𝑥−4 dx
= ln 𝑥 ln 𝑥−
= ln 9 +2 ln 2 − ln ln 1
= ln 9 + ln 4 − ln 8
= ln 9× = ln 9 2

15. WB 13 Show that 𝑒 2 𝑒 4 5𝑥−4 𝑥 2 −𝑥 𝑑𝑥=8+ln 𝑒 4 −1 𝑒 2 −1
5𝑥−4 𝑥 2 −𝑥 = …= 4 𝑥 + 1 𝑥−1
𝑒 2 𝑒 𝑥−4 𝑥 2 −𝑥 𝑑𝑥
= 𝑒 2 𝑒 𝑥 + 1 𝑥−1 𝑑𝑥
= 4 ln 𝑥 + ln 𝑥− 𝑒 4 𝑒 2
= 4 ln 𝑒 4 + ln 𝑒 4 −1 − 4 ln 𝑒 ln (𝑒 2 −1)
= 16 + ln 𝑒 4 −1 −8 − ln (𝑒 2 −1)
= 8+ln 𝑒 4 −1 𝑒 2 −1

16. = 1 𝑥−2 + 2 𝑥+1 + 4 (𝑥+1) 2 𝑑𝑥 = ln 𝑥−2 +2 ln 𝑥+1 −4 𝑥+1 −1 6 3
WB Show that 𝑥+2 (𝑥+3)(𝑥−4) 𝑑𝑥= ln − by rearranging to partial fractions
From WB11 earlier
= 𝑥−2 + 2 𝑥 (𝑥+1) 2 𝑑𝑥
= ln 𝑥−2 +2 ln 𝑥+1 −4 𝑥+1 −
= ln 4 +2 ln 7 − 4 7 − ln 1 +2 ln 4 − 4 4
= ln 4 +2 ln 7 − 4 7 −0−2 ln 4 −1
= 2 ln 7 − ln 4 − = ln − QED

17. Skills 213
homework 213

18. KUS objectives BAT Find integrals using rule for log functions
BAT Solve problems where partial factions used with this rule;
self-assess
One thing learned is –
One thing to improve is –

19. WB 15 Integrate the following:
WB Integrate the following:
Start by trying
y = ln|denominator|
Differentiate
This is double what we want so multiply the ‘guess’ by 1/2

20. END