Question

Evaluate the following:

`int x/(x^4 - 1) "d"x`

Answer 1

Let I = `int x/(x^4 - 1) "d"x`

Put x² = t

⇒ 2x dx = dt

⇒ x dx = `"dt"/2`

`1/2 int "dt"/("t"^2 - 1) = 1/2 int "dt"/("t"^2 - (1)^2)`

= `1/2 * 1/(2 * 1) log |("t" - 1)/("t" + 1)| + "C"`  ....`[because int 1/(x^2 - "a"^2) "d"x = 1/(2"a") log |(x - "a")/(x + "a")| + "C"]`

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

Hence, I = `1/4 log |(x^2 - 1)/(x^2 + 1)| + "C"`