Calculus of a Single Variable, Chapter 8, 8.4, Section 8.4, Problem 28

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