Question

Resolve into a partial fraction for the following:

`1/((x^2 + 4)(x + 1))`

Answer 1

Here the quadratic factor x² + 4 is not factorisable.

Let `1/((x + 1)(x^2 + 4)) = "A"/(x+1) + ("B"x + "C")/(x^2 + 4)`   ....(1)

Multiply both sides by (x + 1) (x² + 4) we get

1 = A(x² + 4) + (Bx + C) (x + 1) ……. (2)

Put x = -1 in (2) we get

1 = A((-1)² + 4) + 0

1 = A(1 + 4)

A = `1/5`

Equating coefficient of x² on both sides of (2) we get,

0 = A + B

0 = `1/5` + B

B = `(-1)/5`

Equating coefficient of x on both sides of (2) we get,

{∵ (Bx + C) (x + 1) = Bx² + Cx = Bx + C}

0 = B + C

0 = `(-1)/5` + C

C = `1/5`

Using A = `1/5,` B = `(-1)/5`, C = `1/5` we get,

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

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

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