Question

Integrate the function:

`(x^2 + x + 1)/((x + 1)^2 (x + 2))`

Answer 1

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`