Question

Integrate the following with respect to x.

`x^3/sqrt(x^8 - 1)`

Answer 1

`int x^3/sqrt(x^8 - 1)  "d"x = int (x^3  "d"x)/sqrt((x^4)^2 - 1^2)`

Let x⁴ = t

Then 4x³dx = dt

= `1/4 int "dt"/sqrt("t"^2 - 1)`

= `1/4 log|"t" + sqrt("t"^2 - 1)| + "c"`

= `1/4 log|x^4 + sqrt(x^8 - 1)| + "c"`