Question

Evaluate:

`int x/((x + 2)(x - 1)^2)dx`

Answer 1

I = `int x/((x + 2)(x - 1)^2)dx`

Let `x/((x + 2)(x - 1)^2) = A/(x + 2) + B/(x - 1) + C/(x - 1)^2`

∴ x = A(x – 1)² + B(x – 1)(x + 2) + C(x + 2)

Put x = 1

1 = 3C,

∴ C = `1/3`

Put x = –2

– 2 = 9A,

∴ A = `-2/9`

Put x = 0

0 = A – 2B + 2C

∴ 2B = `-2/9 + 2/3`

= `(-2 + 6)/9`

= `4/9`

∴ B = `2/9`

∴ I = `int[((-2)/9)/(x + 2) + (2/9)/(x - 1) + (1/3)/(x - 1)^2]dx`

= `-2/9log(x + 2) + 2/9log(x - 1) + 1/3(x - 1)^-1/-1 + c`

= `2/9[log  (x - 1)/(x + 2)] - 1/(3(x - 1)) + c`