int 1/(xsqrt(4x^2+9)) dx Find the indefinite integral

Indefinite integral are written in the form of 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/(xsqrt(4x^2+9)) dx , it resembles one of the formula from integration table.  We may apply the integral formula for rational function with roots as:
int dx/(xsqrt(x^2+a^2))= -1/aln((a+sqrt(x^2+a^2))/x)+C .
 For easier comparison, we  apply u-substitution by letting:  u^2 =4x^2 or (2x)^2 then u = 2x or u/2 =x .
Note: The corresponding value of a^2=9 or 3^2 then  a=3 .
For the derivative of u , we get: du = 2 dx or  (du)/2= dx .
Plug-in the values on the integral problem, we get:
int 1/(xsqrt(4x^2+9)) dx =int 1/((u/2)sqrt(u^2+9)) *(du)/2
                            =int 2/(usqrt(u^2+9)) *(du)/2
                           =int (du)/(usqrt(u^2+9))
Applying the aforementioned integral formula where a^2=9 and a=3 , we get:
int (du)/(usqrt(u^2+9)) =-1/3ln((3+sqrt(u^2+9))/u)+C
Plug-in u^2 =4x^2  and u =2x on  -1/3ln((3+sqrt(u^2+9))/u)+C , we get the indefinite integral as:
int 1/(xsqrt(4x^2+9)) dx=-1/3ln((3+sqrt(4x^2+9))/(2x))+C