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Question
Integrate the rational function:
`x/((x -1)^2 (x+ 2))`
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Solution
Let `x/((x - 1)^2(x + 2))`
`= A/((x - 1)) = B/((x - 1)^2) + C/((x + 2))`
⇒ x = A(x - 1)(x + 2) + B(x + 2) + C(x - 1)2 ...(1)
Put x = 1
1 = 3B
⇒ B = `1/3`
Put x = -2
-2 = C (-2 - 1)2
⇒ C = `(-2)/9`
On comparing the coefficients of x2
A = `-C = 2/9`
Hence, `int x/((x+ 1)^2(x - 2))` dx
`= int 2/ (9 (x - 1)) dx + int 1/ (3 (x - 1)^2) dx - int 2/ (9(x + 2)) dx`
`= 2/9 log abs (x - 1) + 1/3 int (x - 1)^-1/-1 - 2/9 log abs (x + 2) + C`
`= 2/9 log abs ((x - 1)/(x + 2)) - 1/(3(x + 1)) + C`
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