1. Integration By Parts

2. Integration By Parts
When an integral is a product of two functions and neither is the
derivative of the other, we integrate by parts.

3. Integration By Parts
When an integral is a product of two functions and neither is the
derivative of the other, we integrate by parts.
 udv  uv   vdu

4. Integration By Parts
When an integral is a product of two functions and neither is the
derivative of the other, we integrate by parts.
 udv  uv   vdu
u should be chosen so that differentiation makes it a
simpler function.

5. Integration By Parts
When an integral is a product of two functions and neither is the
derivative of the other, we integrate by parts.
 udv  uv   vdu
u should be chosen so that differentiation makes it a
simpler function.
dv should be chosen so that it can be integrated

6. e.g. (i)  x cos xdx

7. e.g. (i)  x cos xdx ux

8. e.g. (i)  x cos xdx ux
du  dx

9. e.g. (i)  x cos xdx ux
du  dx dv  cos xdx

10. e.g. (i)  x cos xdx ux v  sin x
du  dx dv  cos xdx

11. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx

12. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c

13. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c
ii  log xdx

14. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c
ii  log xdx u  log x

15. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c
ii  log xdx u  log x
dx
du 
x

16. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c
ii  log xdx u  log x
dx
du  dv  dx
x

17. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c
ii  log xdx u  log x vx
dx
du  dv  dx
x

18. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c
ii  log xdx u  log x vx
 x log x   dx du 
dx
dv  dx
x

19. e.g. (i)  x cos xdx ux v  sin x
 x sin x   sin xdx du  dx dv  cos xdx
 x sin x  cos x  c
ii  log xdx u  log x vx
 x log x   dx du 
dx
dv  dx
x
 x log x  x  c

20. 1
iii  xe 7 x dx
0

21. 1
iii  xe 7 x dx ux
0

22. 1
iii  xe 7 x dx ux
0
du  dx

23. 1
iii  xe 7 x dx ux
0
du  dx dv  e 7 x dx

24. 1
1 7 x
iii  xe dx
7 x
ux v e
0
7
du  dx dv  e 7 x dx

25. 1
1 7 x
iii  xe dx
7 x
ux v e
0
7
1 1 du  dx dv  e 7 x dx
  xe 7 x    e 7 x dx
1 1
 7
 0 7
 0

26. 1
1 7 x
iii  xe dx
7 x
ux v e
0
7
1 1 du  dx dv  e 7 x dx
  xe 7 x    e 7 x dx
1 1
 7
 0 7
 0
1
 1 7 x 1 7 x 
  xe  e 
 7 49 0

27. 1
1 7 x
iii  xe dx
7 x
ux v e
0
7
1 1 du  dx dv  e 7 x dx
  xe 7 x    e 7 x dx
1 1
 7
 0 7
 0
1
 1 7 x 1 7 x 
  xe  e 
 7 49 0
 1 e 7  1 e 7   0  1 
  
 7 49   49 
8 1
  e 7 
49 49

28. iv  e x cos xdx

29. iv  e x cos xdx u  ex

30. iv  e x cos xdx u  ex
du  e x dx

31. iv  e x cos xdx u  ex
du  e x dx dv  cos xdx

32. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx

33. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x

34. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex

35. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex
du  e x dx

36. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex
du  e x dx dv  sin xdx

37. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex v   cos x
du  e x dx dv  sin xdx

38. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex v   cos x
 e sin x  e cos x   e cos xdx
x x x
du  e x dx dv  sin xdx

39. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex v   cos x
 e sin x  e cos x   e cos xdx
x x x
du  e x dx dv  sin xdx
 2  e x cos xdx  e x sin x  e x cos x

40. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex v   cos x
 e sin x  e cos x   e cos xdx
x x x
du  e x dx dv  sin xdx
 2  e x cos xdx  e x sin x  e x cos x
1 1
 e x cos xdx  e x sin x  e x cos x  c
2 2

41. iv  e x cos xdx u  ex v  sin x
du  e x dx dv  cos xdx
 e sin x   e sin xdx
x x
u  ex v   cos x
 e sin x  e cos x   e cos xdx
x x x
du  e x dx dv  sin xdx
 2  e x cos xdx  e x sin x  e x cos x
1 1
 e x cos xdx  e x sin x  e x cos x  c
2 2
Exercise 2B; 1 to 6 ac, 7 to 11, 12 ace etc