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Question
Evaluate the following:
`int x/(x^4 - 1) "d"x`
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Solution
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"`
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