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Question
Integrate the rational function:
`x/((x + 1)(x+ 2))`
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Solution
Let `x/((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2)`
`=> x/((x + 1)(x + 2)) = (A(x + 2) + B(x + 1))/((x + 1)(x + 2))`
Put x = -1
-1 = A (-1 + 2) ⇒ -1 = A
⇒ A = -1
Put x = -2
-2 = B (-2 + 1) ⇒ -2 = -B
⇒ B = 2
∴ `x/ ((x + 1) (x + 2)) = (-1)/ (x + 1) + 2/ (x + 2)`
∴ `I = int x/ ((x + 1) (x + 2)) dx`
`= int [(-1)/ (x + 1) + 2/ (x + 2)] dx`
`= int (-1)/ ((x + 1)) dx + int 2/ (x + 2) dx`
= - log |x + 1| + 2 log |x + 2| + C
= - log |x + 1| + log |x + 2|2 + C
`= log |((x + 2)^2)/(x + 1)| + C`
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