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Question
Integrate the function:
`(x^2 + x + 1)/((x + 1)^2 (x + 2))`
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Solution
Let `I = (x^2 + x + 1)/((x + 1)^2 (x + 2)) dx`
Now, `(x^2 + x + 1)/((x + 1)^2 (x + 2))`
`= A/(x + 2) + B/(x + 1) + C/(x + 1)^2`
∴ x2 + x + 1 ≡ A(x + 1)2 + B(x + 2)(x + 1) + C(x + 2)
put x = -2.
⇒ 4 - 2 + 1 = A(- 1)2
and 3 = A
A = 3
Put x = - 1
⇒ 1 - 1 + 1 = C(- 1 + 2)
∴ C = 1
Comparing the coefficient of x2,
1 = A + B
B = 1 - A = 1 - 3
∴ B = - 2
∴ `(x^2 + x + 1)/((x + 1)^2 - (x + 2))`
`= 3/(x + 2) - 2/(x + 1) + 1/(x + 1)^2`
∴ I = `int (x^2 + x + 1)/((x + 1)^2 - (x + 2)) dx`
`= int 3/(x + 2) dx - 2int 1/(x + 1) dx + int 1/(x + 1)^2 dx`
`= 3 log (x + 2) - 2 log (x + 1) - 1/(x + 1) + C`
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