1. Integration By
Partial Fractions
Ax
To find; dx
P x
2. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
3. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
4. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
A
a) for linear factor x a , write
xa
5. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
A
a) for linear factor x a , write
xa
b) for multiple linear factors x a , write
n
A B C
x a x a 2
x a n
6. Integration By
Partial Fractions
Ax
To find; dx
P x
1 If degA x degP x , perform a division
2 If degA x degP x , factorise P x
A
a) for linear factor x a , write
xa
b) for multiple linear factors x a , write
n
A B C
x a x a 2
x a n
Ax B
c) for polynomial factors e.g. ax bx c, write 2
2
ax bx c
8. x2 x
e.g. i dx x 1 x2 0x 0
x 1
x2 x
9. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x0
10. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x0
x 1
11. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x0
x 1
1
12. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
x 1
1
13. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
14. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx
ii
x2 x
15. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx
ii
x2 x
3dx
x x 1
16. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii 2
x x x x 1 x x 1
3dx
x x 1
17. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
18. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
x0
A3
A 3
19. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
x0 x 1
A3 B3
A 3
20. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
3 3 x0 x 1
dx A3 B3
x x 1
A 3
21. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
3 3 x0 x 1
dx A3 B3
x x 1
3 log x 3 log x 1 c A 3
22. x2 x 1
e.g. i dx x 1 x2 0x 0
x 1
x2 x
x 1
1
dx
x 1 x0
1 x 1
x 2 x log x 1 c
2 1
3dx A B 3
ii
x2 x x x 1 x x 1
3dx A x 1 Bx 3
x x 1
3 3 x0 x 1
dx A3 B3
x x 1
3 log x 3 log x 1 c A 3
x 1
3 log
c
x
24. x5
iii 2 dx
x 3 x 10
x5
dx
x 5 x 2
25. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2
26. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2
7B 3
3
B
7
27. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
7B 3 7 A 10
3 10
B A
7 7
28. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
B A
7 7
29. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
30. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
dx
iv 3
x x
31. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
dx
iv 3
x x
dx
xx 2 1
32. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1
33. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1
x0
A 1
34. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
A 1 B Ci 1
35. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
A 1 B Ci 1
B 1 C 0
36. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
1 x dx
A 1 B Ci 1
x x 1
2
B 1 C 0
37. x5 x5
iii 2 dx A
B
x 3 x 10 x 5 x 2 x 5 x 2
x5
dx A x 2 B x 5 x 5
x 5 x 2 x 2 x5
10 3 7B 3 7 A 10
dx
7 x 5 7 x 2 3 10
10 3 B A
log x 5 log x 2 c 7 7
7 7
A Bx C 1
iv 3
dx 2
x x x x 1 xx 2 1
dx Ax 2 1 Bx C x 1
xx 2 1 xi
x0
1 x dx
A 1 B Ci 1
x x 1
2
B 1 C 0
log x logx 2 1 c
1
2
39. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
40. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1
2 B 1
1
B
2
41. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1 xi
2 B 1 2C 2 Di i
1
B
2
42. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1 xi
2 B 1 2C 2 Di i
1
1 C 0 D
B 2
2
43. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
x 1 xi
2 B 1 2C 2 Di i
1
1 C 0 D
B 2
2
x0
2A B D 0
1 1
2A 0
2 2
A0
44. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
B 2
2
x0
2A B D 0
1 1
2A 0
2 2
A0
45. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1
dx 2
x0
2A B D 0
1 1
2A 0
2 2
A0
46. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1
dx 2
x0
1 x 11 1 2A B D 0
tan x c
2 1 1 1
2A 0
2 2
A0
47. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1
dx 2
x0
1 x 11 1 2A B D 0
tan x c
2 1 1 1
2A 0
1 1 2 2
tan 1 x c A0
2 x 1
48. v
xdx A B Cx D x
2
x 12 x 2 1 x 1 x 12 x 1 x 12 x 2 1
A x 1x 2 1 B x 2 1 Cx D x 1 x
2
1 x 1 xi
1
2 x 1 2x 1
2 2 dx 2 B 1 2C 2 Di i
1
1 C 0 D
1 1 B 2
x 1 2
2
2 x 1
dx 2
x0
1 x 11 1 2A B D 0
tan x c
2 1 1 1
2A 0
1 1 Exercise 2G; 2 2
tan 1 x c 1, 3, 5, 7 to 21 A0
2 x 1