Question

Integrate the rational function:

`(3x -1)/(x + 2)^2`

Answer 1

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`