Question

`int ("d"x)/(x(x^4 + 1))` = ______.

Options

• `1/4 log |(x^4 + 1)/x^4| + "c"`

• `1/4 log |x^4/(1 + x^4)| + "c"`

• `1/4 log|x^4 + 1| + "c"`

• `1/4 log |x^4/(x^4 + 2)| + "c"`

Answer 1

`int ("d"x)/(x(x^4 + 1))` = `1/4 log |x^4/(1 + x^4)| + "c"`.

Explanation:

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

 `int x^3/(x^4(x^4 + 1)) "d"x`

Put x⁴ = t ⇒ 4x³ dx = dt

∴ I = `1/4 int "dt"/("t"(1 + "t"))`

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

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

= `1/4 log|"t"/(1 + "t")| + "c"`

= `1/4 log|x^4/(1 + x^4)| + "c"`