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

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/(xsqrt(9x^2+1)) 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 =9x^2 or (3x)^2 then u = 3x or u/3 =x .
Note: The corresponding value of a^2=1 or 1^2 then a=1 .
For the derivative of u , we get: du = 3 dx or (du)/3= dx .
Plug-in the values on the integral problem, we get:
int 1/(xsqrt(9x^2+1)) dx =int 1/((u/3)sqrt(u^2+1)) *(du)/3
                         =int 3/(usqrt(u^2+1)) *(du)/3
                         =int (du)/(usqrt(u^2+1))
Applying the aforementioned integral formula where a^2=1 and a=1 , we get:
int (du)/(usqrt(u^2+1)) =-1/1ln((1+sqrt(u^2+1))/u)+C
                  =-ln((1+sqrt(u^2+1))/u)+C
                  =ln(((1+sqrt(u^2+1))/u)^-1) + C
                  =ln(u/(1+sqrt(u^2+1))) + C
Plug-in u^2 =9x^2  and u =3x and we get the indefinite integral as:
int 1/(xsqrt(9x^2+1)) dx=ln((3x)/(1+sqrt(9x^2+1)))+C

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