Question

How do you integrate `(x^2-2)/((x+1)(x^2+3))` using partial fractions?

Answer 1

Please see the explanation.

Explanation:

Expand:

`(x^2 - 2)/((x + 1)(x^2 + 3)) = A/(x + 1) + (Bx + C)/(x^2 + 3)`

`x^2 - 2 = A(x^2 + 3) + (Bx + C)(x + 1)`

Let x = -1 to make B and C disappear:

`-1^2 - 2 = A(-1^2 + 3)`

`-1 = 4A`

`A = -1/4`

`x^2 - 2 = -1/4(x^2 + 3) + (Bx + C)(x + 1)`

Let x = 0 to make B disappear:

`- 2 = -1/4(3) + (C)(1)`

`C = -5/4`

`x^2 - 2 = -1/4(x^2 + 3) + (Bx -5/4)(x + 1)`

Let x = 1:

`1^2 - 2 = -1/4(1^2 + 3) + (B -5/4)(1 + 1)`

`-1 = -1 + (2B -5/2)`

`B = 5/4`

`int(x^2 - 2)/((x + 1)(x^2 + 3))dx = -1/4int1/(x + 1)dx + 5/4intx/(x^2 + 3)dx - 5/4int1/(x^2 + 3)dx`

`int(x^2 - 2)/((x + 1)(x^2 + 3))dx = -1/4ln|x + 1| + 5/4ln|x^2 + 3| - (5sqrt(3))/12tan^-1(x/sqrt(3))`