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
10. e.g. (i) x cos xdx ux v sin x
du dx dv cos xdx
11. e.g. (i) x cos xdx ux v sin x
x sin x sin xdx du dx dv cos xdx
12. e.g. (i) x cos xdx ux 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 ux 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 ux 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 ux 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 ux 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 ux 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 vx
dx
du dv dx
x
18. e.g. (i) x cos xdx ux 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 vx
x log x dx du
dx
dv dx
x
19. e.g. (i) x cos xdx ux 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 vx
x log x dx du
dx
dv dx
x
x log x x c
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