Calculus of a Single Variable, Chapter 8, 8.4, Section 8.4, Problem 27

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