Question

Integrate the rational function:

`1/(x^4 - 1)`

Answer 1

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

`= A/(x + 1) + B/(x - 1) + (Cx + D)/(x^2 + 1)`

1 ≡  A(x – 1) (x² + 1) + B(x + 1) (x² + 1) + (Cx + D) (x + 1) (x – 1)     …(1)

Putting x = -1 in equation (1),

1 = A (-1 – 1) (1 + 1)

⇒ 1 = A (-4)

⇒ A = `-1/4`

Putting x = 1 in equation (1),

1 = B (1 + 1) (1 + 1)

⇒ 1= B (2) (2)

⇒ B = `1/4`

Comparing the coefficients of x³ in equation (1),

0 = A + B + C

`=> 0 = (-1)/4 + 1/4 + C`

⇒  C = 0

1 = -A + B - D

`=> 1 = 1/4 + 1/4 - D`

⇒ ` D = -1/2`

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

`therefore int  dx/(x^4 - 1) = 1/4 int 1/(x + 1)  dx + 1/4 int 1/(x - 1)  dx - 1/2 int 1/(x^2 + 1)  dx`

`= - 1/4  log (x + 1) = 1/4  log (x - 1) -1/2  tan^-1 x + C`

`= 1/4  log ((x - 1)/(x + 1)) - 1/2  tan^-1 x + C`