Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 19

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