1. Integrating Derivative
on Function

2. Integrating Derivative
on Function

f  x 
dx  log f  x   c
f x

3. Integrating Derivative
on Function

e.g. (i)
1
 7  3x dx
f  x 
dx  log f  x   c
f x

4. Integrating Derivative
on Function

e.g. (i)
1
 7  3x dx
1 3
 
dx
3 7  3x
f  x 
dx  log f  x   c
f x

5. Integrating Derivative
on Function

e.g. (i)
f  x 
dx  log f  x   c
f x
1
 7  3x dx
1 3
 
dx
3 7  3x
1
  log7  3 x   c
3

6. Integrating Derivative
on Function

e.g. (i)
f  x 
dx  log f  x   c
f x
1
 7  3x dx
1 3
 
dx
3 7  3x
1
  log7  3 x   c
3
ii 
dx
 8x  5

7. Integrating Derivative
on Function

e.g. (i)
f  x 
dx  log f  x   c
f x
1
 7  3x dx
1 3
 
dx
3 7  3x
1
  log7  3 x   c
3
dx
 8x  5
1 8dx
 
8 8x  5
ii 

8. Integrating Derivative
on Function

e.g. (i)
f  x 
dx  log f  x   c
f x
1
 7  3x dx
1 3
 
dx
3 7  3x
1
  log7  3 x   c
3
dx
 8x  5
1 8dx
 
8 8x  5
1
 log8 x  5  c
8
ii 

9. Integrating Derivative
on Function

f  x 
dx  log f  x   c
f x
e.g. (i)  1 dx
7  3x
1 3
 
dx
3 7  3x
1
  log7  3 x   c
3
dx
ii  
8x  5
1 8dx
 
8 8x  5
1
 log8 x  5  c
8
x5
iii   6 dx
x 2

10. Integrating Derivative
on Function

f  x 
dx  log f  x   c
f x
e.g. (i)  1 dx
7  3x
1 3
 
dx
3 7  3x
1
  log7  3 x   c
3
dx
ii  
8x  5
1 8dx
 
8 8x  5
1
 log8 x  5  c
8
x5
iii   6 dx
x 2
1 6 x5
  6
dx
6 x 2

11. Integrating Derivative
on Function

f  x 
dx  log f  x   c
f x
e.g. (i)  1 dx
7  3x
1 3
 
dx
3 7  3x
1
  log7  3 x   c
3
dx
ii  
8x  5
1 8dx
 
8 8x  5
1
 log8 x  5  c
8
x5
iii   6 dx
x 2
1 6 x5
  6
dx
6 x 2
1
 logx 6  2   c
6

12. 1
iv   dx
5x

13. 1
iv   dx
5x
1 5
  dx
5 5x

14. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5

15. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
OR
1 1
 x dx
5

16. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
OR
1 1
 x dx
5
1
 log x  c
5

17. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
4x 1
v  
dx
2x 1
OR
1 1
 x dx
5
1
 log x  c
5

18. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
4x 1
v  
dx
2x 1
OR
1 1
 x dx
5
1
 log x  c
5
order numerator  order denominator

19. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
4x 1
v  
dx
2x 1
OR
1 1
 x dx
5
1
 log x  c
5
order numerator  order denominator
 polynomial division

20. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
4x 1
v  
dx
2x 1
OR
1 1
 x dx
5
1
 log x  c
5
order numerator  order denominator
 polynomial division
2
2x 1 4x 1
4x  2
1

21. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
4x 1
dx
v  
2x 1
2  1  dx
 

 2 x  1
OR
1 1
 x dx
5
1
 log x  c
5
order numerator  order denominator
 polynomial division
2
2x 1 4x 1
4x  2
1

22. 1
iv   dx
5x
1 5
  dx
5 5x
1
 log 5 x  c
5
OR
1 1
 x dx
5
1
 log x  c
5
order numerator  order denominator
4x 1
dx
v  
 polynomial division
2x 1
2  1  dx
2
 
2x 1 4x 1

 2 x  1
1
4x  2
 2 x  log2 x  1  c
2
1

23. vi 
2
2x
 x 2  1dx
1

24. vi 
2
2x
 x 2  1dx
1
 logx  11
2
2

25. vi 
2
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2

26. vi 
2
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2

27. vi 
2
vii  Differentiate x 3 log x and hence
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2
integrate x 2 log x

28. vi 
2
vii  Differentiate x 3 log x and hence
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2
integrate x 2 log x
d 3
3 1
x log x  x    log x 3x 2 
dx
 x

29. vi 
2
vii  Differentiate x 3 log x and hence
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2
integrate x 2 log x
d 3
3 1
x log x  x    log x 3x 2 
dx
 x
 x 2  3 x 2 log x

30. vi 
2
vii  Differentiate x 3 log x and hence
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2
integrate x 2 log x
d 3
3 1
x log x  x    log x 3x 2 
dx
 x
 x 2  3 x 2 log x
 x
2
 3 x 2 log x dx  x 3 log x  c

31. vi 
2
vii  Differentiate x 3 log x and hence
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2
integrate x 2 log x
d 3
3 1
x log x  x    log x 3x 2 
dx
 x
 x 2  3 x 2 log x
 x
2
 3 x 2 log x dx  x 3 log x  c
3 x 2 log xdx  x 3 log x   x 2 dx  c

32. vi 
2
vii  Differentiate x 3 log x and hence
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2
integrate x 2 log x
d 3
3 1
x log x  x    log x 3x 2 
dx
 x
 x 2  3 x 2 log x
 x
2
 3 x 2 log x dx  x 3 log x  c
3 x 2 log xdx  x 3 log x   x 2 dx  c
1 3
1 3
 x log xdx  3 x log x  9 x  c
2

33. vi 
2
vii  Differentiate x 3 log x and hence
2x
 x 2  1dx
1
 logx  11
2
2
 log 5  log 2
5
 log 
2
integrate x 2 log x
d 3
3 1
x log x  x    log x 3x 2 
dx
 x
 x 2  3 x 2 log x
 x
2
 3 x 2 log x dx  x 3 log x  c
3 x 2 log xdx  x 3 log x   x 2 dx  c
1 3
1 3
 x log xdx  3 x log x  9 x  c
2
Exercise 12D; 1 to 12 ace in all, 14a*
Exercise 12E; 1 to 6 all, 7 to 21 odds, 22abc*, 23*