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Question
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Solution
Let `(3x - 1)/((x - 1)(x - 2)(x - 3))`
`= A/(x - 1) + B/(x - 2) + C/(x - 3)`
⇒ 3x - 1 = A(x - 2) (x - 3) + B(x - 1) (x - 3) + C(x - 1) (x - 2) …(1)
Putting x = 1 in (i), we get
3 - 1 = A(1 - 2) (1 - 3)
⇒ 2 = A(-1) (-2)
⇒ A = 1
Putting x = 2 in (i), we get
6 - 1 = B (2 - 1) (2 - 3)
⇒ 5 = B(1) (-1)
⇒ B = -5
Putting x = 3 in (i), we get
9 - 1 = C (3 - 1) (3 - 2)
⇒ 8 = C (2) (1)
⇒ C = 4
`therefore (3x - 1)/((x - 1)(x - 2)(x - 3))`
`= 1/(x - 1) - 5/(x - 2) + 4/(x - 3)`
`= int (3x - 1)/((x - 1)(x - 2)(x - 3))` dx
`= int1/(x - 1) dx - 5 int 1/(x - 2) dx + 4 int 1/(x - 3) dx`
= log (x - 1) - 5 log (x - 2) + 4 log (x - 3) + C
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