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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
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1. In this section we will show how to integrate
rational functions, that is, functions of the form
P(x)
Q(x)
where P and Q are polynomials.
Integrals of Rational Functions
2. In this section we will show how to integrate
rational functions, that is, functions of the form
P(x)
Q(x)
where P and Q are polynomials.
Rational Decomposition Theorem
Given reduced P/Q where deg P < deg Q,
then P/Q = F1 + F2 + .. + Fn where
Fi = orA
(ax + b)k
Ax + B
(ax2 + bx + c)k
and that (ax + b)k or (ax2 + bx + c)k are
factors of Q(x).
Integrals of Rational Functions
3. Therefore finding the integrals of rational
functions is reduced to finding integrals of
orA
(ax + b)k
Ax + B
(ax2 + bx + c)k
Integrals of Rational Functions
4. Therefore finding the integrals of rational
functions is reduced to finding integrals of
orA
(ax + b)k
Ax + B
(ax2 + bx + c)k
Integrals of Rational Functions
The integrals ∫ are straight forward
with the substitution method by setting
u = ax + b.
dx
(ax + b)k
5. Therefore finding the integrals of rational
functions is reduced to finding integrals of
orA
(ax + b)k
Ax + B
(ax2 + bx + c)k
Integrals of Rational Functions
The integrals ∫ are straight forward
with the substitution method by setting
u = ax + b.
dx
(ax + b)k
To integrate ∫ dx, we need to
complete the square of the denominator.
Ax + B
(ax2 + bx + c)k
7. Complete the square of x2 + 6x + 10 – it’s irreducible.
Integrals of Rational Functions
x
x2 + 6x + 10Example A. Find ∫ dx
8. x2 + 6x + 10 = (x2 + 6x ) + 10
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
9. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
10. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
= (x + 3)2 + 1
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
11. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
= (x + 3)2 + 1
x
x2 + 6x + 10∫ dx
= ∫
x
(x + 3)2 + 1
dx
Hence
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
12. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
= (x + 3)2 + 1
x
x2 + 6x + 10∫ dx
Set u = x + 3
= ∫
x
(x + 3)2 + 1
dx
substitution
Hence
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
13. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
= (x + 3)2 + 1
x
x2 + 6x + 10∫ dx
Set u = x + 3
x = u – 3
dx = du= ∫
x
(x + 3)2 + 1
dx
substitution
Hence
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
14. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
= (x + 3)2 + 1
x
x2 + 6x + 10∫ dx
Set u = x + 3
x = u – 3
dx = du= ∫
x
(x + 3)2 + 1
dx
substitution
= ∫
u – 3
u2 + 1
du
Hence
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
15. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
= (x + 3)2 + 1
x
x2 + 6x + 10∫ dx
Set u = x + 3
x = u – 3
dx = du= ∫
x
(x + 3)2 + 1
dx
substitution
= ∫
u – 3
u2 + 1
du
= ∫
u
u2 + 1 du – 3 ∫
1
u2 + 1 du
Hence
Integrals of Rational Functions
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
16. x2 + 6x + 10 = (x2 + 6x + 9) + 10 – 9
= (x + 3)2 + 1
x
x2 + 6x + 10∫ dx
Set u = x + 3
x = u – 3
dx = du= ∫
x
(x + 3)2 + 1
dx
substitution
= ∫
u – 3
u2 + 1
du
= ∫
u
u2 + 1 du – 3 ∫
1
u2 + 1 du
Hence
Integrals of Rational Functions
Set w = u2 + 1
substitution
Complete the square of x2 + 6x + 10 – it’s irreducible.
x
x2 + 6x + 10Example A. Find ∫ dx
17. Set w = u2 + 1
= 2u
substitution
dw
du
Integrals of Rational Functions
= ∫
u
u2 + 1 du – 3 ∫
1
u2 + 1 du Set w = u2 + 1
18. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
Integrals of Rational Functions
= ∫
u
u2 + 1 du – 3 ∫
1
u2 + 1 du Set w = u2 + 1
19. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
= ∫
u
w
– 3 ∫
1
u2 + 1 du
dw
2u
Integrals of Rational Functions
= ∫
u
u2 + 1 du – 3 ∫
1
u2 + 1 du Set w = u2 + 1
20. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
= ∫
u
w
– 3 ∫
1
u2 + 1 du
dw
2u
= ½ ∫ 1
w – 3 tan-1(u) + cdw
Integrals of Rational Functions
21. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
= ∫
u
w
– 3 ∫
1
u2 + 1 du
dw
2u
= ½ ∫ 1
w – 3 tan-1(u) + cdw
= ½ Ln(w) – 3 tan-1(x + 3) + c
Integrals of Rational Functions
22. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
= ∫
u
w
– 3 ∫
1
u2 + 1 du
dw
2u
= ½ ∫ 1
w – 3 tan-1(u) + cdw
= ½ Ln(lwl) – 3 tan-1(x + 3) + c
= ½ Ln((x + 3)2 + 1) – 3 tan-1(x + 3) + c
Integrals of Rational Functions
23. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
= ∫
u
w
– 3 ∫
1
u2 + 1 du
dw
2u
= ½ ∫ 1
w – 3 tan-1(u) + cdw
= ½ Ln(lwl) – 3 tan-1(x + 3) + c
= ½ Ln((x + 3)2 + 1) – 3 tan-1(x + 3) + c
Integrals of Rational Functions
3x
x2 + 7x + 10Example: Find ∫ dx
24. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
= ∫
u
w
– 3 ∫
1
u2 + 1 du
dw
2u
= ½ ∫ 1
w – 3 tan-1(u) + cdw
= ½ Ln(lwl) – 3 tan-1(x + 3) + c
= ½ Ln((x + 3)2 + 1) – 3 tan-1(x + 3) + c
Integrals of Rational Functions
Since x2 + 7x + 10 = (x + 2)(x + 5), we can
decompose the rational expression.
3x
x2 + 7x + 10Example: Find ∫ dx
25. Set w = u2 + 1
= 2u
substitution
dw
du
du =
dw
2u
= ∫
u
w
– 3 ∫
1
u2 + 1 du
dw
2u
= ½ ∫ 1
w – 3 tan-1(u) + cdw
= ½ Ln(lwl) – 3 tan-1(x + 3) + c
= ½ Ln((x + 3)2 + 1) – 3 tan-1(x + 3) + c
Integrals of Rational Functions
3x
x2 + 7x + 10Example: Find ∫ dx
Since x2 + 7x + 10 = (x + 2)(x + 5), we can
decompose the rational expression.
Specifically 3x
(x + 2)(x + 5) =
A
(x + 2)
+ B
(x + 5)
26. Integrals of Rational Functions
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
28. Integrals of Rational Functions
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
29. Integrals of Rational Functions
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B so B = 5
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
30. Integrals of Rational Functions
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B so B = 5
Evaluate at x = -2, we get -6 = 3A
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
31. Integrals of Rational Functions
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B so B = 5
Evaluate at x = -2, we get -6 = 3A so A = -2
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
32. Integrals of Rational Functions
Hence 3x
(x + 2)(x + 5) =
-2
(x + 2)
+ 5
(x + 5)
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B so B = 5
Evaluate at x = -2, we get -6 = 3A so A = -2
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
33. Integrals of Rational Functions
Hence x
(x + 2)(x + 5) =
-2
(x + 2)
+ 5
(x + 5)
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B so B = 5
Evaluate at x = -2, we get -6 = 3A so A = -2
3x
x2 + 7x + 10So ∫ dx
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
34. Integrals of Rational Functions
Hence x
(x + 2)(x + 5) =
-2
(x + 2)
+ 5
(x + 5)
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B so B = 5
Evaluate at x = -2, we get -6 = 3A so A = -2
3x
x2 + 7x + 10So ∫ dx
= ∫ dx
-2
(x + 2)
+ 5
(x + 5)
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
35. Integrals of Rational Functions
Hence x
(x + 2)(x + 5) =
-2
(x + 2)
+ 5
(x + 5)
3x = A(x + 5) + B(x + 2)
Evaluate at x = -5, we get -15 = -3B so B = 5
Evaluate at x = -2, we get -6 = 3A so A = -2
3x
x2 + 7x + 10So ∫ dx
= -2Ln(lx + 2l) + 5Ln(lx + 5l) + c
= ∫ dx
-2
(x + 2)
+ 5
(x + 5)
Clear denominators: 3x
(x + 2)(x + 5) =
A
(x + 2)
+
B
(x + 5)
37. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
1
(x2 + 1)2xExample: Find ∫ dx
38. Integrals of Rational Functions
1
(x2 + 1)2xExample: Find ∫ dx
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
39. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
1
(x2 + 1)2xExample: Find ∫ dx
40. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
1
(x2 + 1)2xExample: Find ∫ dx
41. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
1
(x2 + 1)2xExample: Find ∫ dx
42. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
There is no x4-term,
1
(x2 + 1)2xExample: Find ∫ dx
43. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
There is no x4-term, hence Ax4 + x4 = 0,
1
(x2 + 1)2xExample: Find ∫ dx
44. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
There is no x4-term, hence Ax4 + x4 = 0, so A = -1
1
(x2 + 1)2xExample: Find ∫ dx
45. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
There is no x4-term, hence Ax4 + x4 = 0, so A = -1
There is no x3-term,
1
(x2 + 1)2xExample: Find ∫ dx
46. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
There is no x4-term, hence Ax4 + x4 = 0, so A = -1
There is no x3-term, hence Bx3 = 0,
1
(x2 + 1)2xExample: Find ∫ dx
47. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
There is no x4-term, hence Ax4 + x4 = 0, so A = -1
There is no x3-term, hence Bx3 = 0, so B = 0
1
(x2 + 1)2xExample: Find ∫ dx
48. Integrals of Rational Functions
1
(x2 + 1)2x =
Ax + B
(x2 + 1)
+ Cx + D
(x2 + 1)2 + E
x
Clear denominators
1 = (Ax + B)(x2 + 1)x +(Cx + D) x +E(x2 + 1)2
Evaluate this at x = 0, we get 1 = E
Hence 1 = (Ax + B)(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
There is no x4-term, hence Ax4 + x4 = 0, so A = -1
There is no x3-term, hence Bx3 = 0, so B = 0
So we've1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
1
(x2 + 1)2xExample: Find ∫ dx
49. Integrals of Rational Functions
There is no x-term in
1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
50. Integrals of Rational Functions
There is no x-term in
1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
Hence Dx = 0, or D = 0.
51. Integrals of Rational Functions
There is no x-term in
1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
Hence Dx = 0, or D = 0. So the expression is
1 = -x2
(x2 + 1)+Cx2 +1(x2 + 1)2
52. Integrals of Rational Functions
There is no x-term in
1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
Hence Dx = 0, or D = 0. So the expression is
1 = -x2
(x2 + 1)+Cx2 +1(x2 + 1)2
Expand this
53. Integrals of Rational Functions
There is no x-term in
1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
Hence Dx = 0, or D = 0. So the expression is
1 = -x2
(x2 + 1)+Cx2 +1(x2 + 1)2
Expand this
1 = -x4 – x2
Cx2 + x4 + 2x2 + 1+
54. Integrals of Rational Functions
There is no x-term in
1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
Hence Dx = 0, or D = 0. So the expression is
1 = -x2
(x2 + 1)+Cx2 +1(x2 + 1)2
Expand this
1 = -x4 – x2
Cx2 + x4 + 2x2 + 1+
Hence Cx2 + x2 = 0 or C = -1
55. Integrals of Rational Functions
There is no x-term in
1 = -x(x2 + 1)x +(Cx + D)x +1(x2 + 1)2
Hence Dx = 0, or D = 0. So the expression is
1 = -x2
(x2 + 1)+Cx2 +1(x2 + 1)2
Expand this
1 = -x4 – x2
Cx2 + x4 + 2x2 + 1+
Hence Cx2 + x2 = 0 or C = -1
Put it all together
1
(x2 + 1)2x =
-x
(x2 + 1)
+ -x
(x2 + 1)2 + 1
x
61. Integrals of Rational Functions
1
(x2 + 1)2xTherefore ∫ dx
= ∫
-x dx
(x2 + 1)
+ -x dx
(x2 + 1)2 + dx
x∫ ∫
substitution
set u = x2 + 1
du
dx = 2x
du
2xdx == ∫
-x
u
du
2x
+ ∫
-x
u2
du
2x
+ Ln(lxl)
62. Integrals of Rational Functions
1
(x2 + 1)2xTherefore ∫ dx
= ∫
-x dx
(x2 + 1)
+ -x dx
(x2 + 1)2 + dx
x∫ ∫
substitution
set u = x2 + 1
du
dx = 2x
du
2xdx == ∫
-x
u
du
2x
+ ∫
-x
u2
du
2x
+ Ln(lxl)
= - ½ ∫
du
u + Ln(lxl)½ ∫
du
u2–
63. Integrals of Rational Functions
1
(x2 + 1)2xTherefore ∫ dx
= ∫
-x dx
(x2 + 1)
+ -x dx
(x2 + 1)2 + dx
x∫ ∫
substitution
set u = x2 + 1
du
dx = 2x
du
2xdx == ∫
-x
u
du
2x
+ ∫
-x
u2
du
2x
+ Ln(x)
= - ½ ∫
du
u + Ln(x)½ ∫
du
u2–
= - ½ Ln(u) + ½ u-1 + Ln(x) + c
64. Integrals of Rational Functions
1
(x2 + 1)2xTherefore ∫ dx
= ∫
-x dx
(x2 + 1)
+ -x dx
(x2 + 1)2 + dx
x∫ ∫
substitution
set u = x2 + 1
du
dx = 2x
du
2xdx == ∫
-x
u
du
2x
+ ∫
-x
u2
du
2x
+ Ln(x)
= - ½ ∫
du
u + Ln(x)½ ∫
du
u2–
= - ½ Ln(u) + ½ u-1 + Ln(x) + c
= - ½ Ln(x2 + 1) + + Ln(x) + c1
2(x2 + 1)