Recall that indefinite integral follows the formula: `int f(x) dx = F(x) +C`

where: `f(x)` as the integrand

`F(x)` as the anti-derivative function

`C` as the arbitrary constant known as constant of integration

For the given problem `int 1/(x^2sqrt(x^2-4)) dx` , it resembles one of the formula from integration table. We may apply the integral formula for rational function with roots as:

`int 1/(u^2sqrt(u^2-a^2))du = 1/(a^2*u) sqrt(u^2-a^2)+C` .

By comparing "`u^2-a^2` " with "`x^2-4` " , we determine the corresponding values as:

`u^2=x^2` then `u =x`

`a^2 =4`

Plug-in the values on the aforementioned integral formula for rational function with roots where `a^2 =4` , we get:

`int 1/(x^2sqrt(x^2-4)) dx=1/(4*x) sqrt(x^2-4)+C`

` =1/(4x) sqrt(x^2-4)+C`