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