Change the fraction into partial fractions, using denominators that are factors of the fraction.
_____1
=
1
= A +
B
x2 - 5x +
6
(x - 2)(x - 3) x -
2
x - 3
Multiply both sides by the common denominator.
1 = A(x - 3) + B(x - 2)
Choose random values to plug in for x to get expressions in A and B.
Let x =
3
Let x = 2
1 =
B
1 = -A
-1 = A
So _____1
= -1
+ 1
x2 - 5x +
6
x - 2 x - 3
Integrate:
5x2 + 20x + 6
= A +
B +
C
x(x + 1)2
x x +
1
(x + 1)2
5x2 + 20x + 6 = A(x + 1)2 + Bx(x + 1) + Cx
Let x =
-1
Let x =
0
Let x = 1
-9=
-C
6 =
A
31 = 4A + 2B + C
9=
C
31 = 4(6) + 2B + 9
31 = 24 + 2B + 9
31 = 33 + 2B
-2 = 2B
-1 = B
= 6 ln |x| -
ln
|x + 1| -
9
+ C
x + 1
2x3 - 4x -
8
= A
+
B
+
Cx + D
x(x - 1)(x2 +
4)
x
x -
1
x2 + 4
2x3 - 4x - 8 = A(x - 1)(x2 + 4) + Bx(x2 + 4) + (Cx + D)x(x - 1)
Let x =
1
Let x =
0
Let x =
-1
Let x = 2
-10 =
5B
-8 =
-4A
-6 = -10A - 5B - 2C +
2D
0 = 8A + 16B + 4C + 2D
-2 =
B
2 =
A
-6 = -10(2) - 5(-2) - 2C + 2D 0 = 8(2) + 16(-2)
+ 4C + 2D
-6 = -20 + 10 - 2C +
2D
0 = 16 - 32 + 4C + 2D
-6 = -10 - 2C +
2D
0 = -16 + 4C + 2D
4 = -2C +
2D
16 = 4C + 2D
2 = -C +
D
8 = 2C + D
Now combine the two equations in two unknowns:
8 = 2C + D
2 = -C + D
Subtract:
6 = C
Substitute C back into one of the sentences: 2 = -6 + D
8 = D
= 2 ln |x| + 2 ln |x
- 1| + 3 ln (x2 +
4)
+ 4 arctan x/2 + C