Question

Integrate the rational function:

`1/(x(x^4 - 1))`

Answer 1

Let `I = int 1/ (x (x^4 - 1))  dx`

`= 1/4 int (4x^3)/(x^4(x^4 - 1))  dx`

Put x⁴ = t

⇒ 4x³ dx = dt

∴ `I = 1/4 int dt/(t(t - 1))`

Let `1/(t (t - 1)) = A/t + B/(t - 1)`

⇒ 1 = A (t - 1) + Bt               ....(i)

Putting t = 0 in (i), we get

1 = A (-1)

⇒ A = -1

Putting t = 1 in (i), we get

1 = B (1)

⇒ B = 1

∴ `1/ (t (t - 1)) = (-1)/t + 1/ (t - 1)`

∴ `I = 1/4 int (-1/t + 1/ (t - 1))  dt`

`= 1/4 [-log |t| + log |t - 1|] + C`

`= 1/4 log |(t - 1)/t| + C`

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