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Question
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Solution
Let `I = int (3x - 1)/ (x + 2)^2 dx`
Let `(3x - 1)/(x + 2)^2 = A/(x + 2) + B/ (x + 2)^2`
⇒ 3x - 1 = A (x + 2) + B .....(i)
Comparing coefficients of x in (i), we get
A = 3
Comparing the coefficients of constant terms in (i), we get
2A + B = -1
Put A = 3 in (ii), and we get 6 + B = -1
⇒ B = -7
∴ `(3x - 1)/(x + 2)^2 = 3/ (x + 2) + (-7)/(x + 2)^2`
⇒ `int (3x - 1)/(x + 2)^2 dx = 3 int dx/ (x + 2) - 7 int dx/ (x + 2)^2`
`= 3 log |x + 2| -7 (x + 2)^-1/-1 + C`
`= 3 log |x + 2| + 7/ (x + 2) + C`
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