Question

Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`

Answer 1

Let I = `int (1)/(x(x^3 - 1)).dx`

= `int (x^-4)/(x^-4x(x^3 - 1)).dx`

= `int (x^-4)/(1 - x^-3).dx` 

= `(1)/(3) int (3x^-4)/(1 - x^-3).dx`

= `(1)/(3) int (d/dx(1 - x^-3))/(1 - x^-3).dx`

= `(1)/(3)log|1 - x^-3 | + c       ...[∵ int (f'(x))/f(x)dx = log|f(x)| + c]`

= `(1)/(3)log|1 - 1/x^3|  + c`

= `(1)/(3)log|(x^3 - 1)/x^3| + c`.

Alternative Method :

Let I = `int (1)/(x(x^3 - 1)).dx`

= `int x^2/(x^3(x^3 - 1)).dx`

Put x³ = t
∴ 3x²dx = dt

∴ x²dx = `dt/(3)`

∴ I = `int (1)/(t(t -  1)).dt/(3)`

= `(1)/(3)int(1)/(t(t - 1))dt`

= `(1)/(3) int(t - (t - 1))/(t(t - 1))dt` 

= `(1)/(3) int(1/(t - 1) - 1/t)dt`

= `(1)/(3)[int (1)/(t - 1)dt - int (1)/tdt]`

= `(1)/(3)[log |t - 1| - log|t|] + c`

= `(1)/(3)log|(t - 1)/t| + c`

= `(1)/(3)log|(x^3 - 1)/x^3| + c`.