Calculus of a Single Variable, Chapter 8, 8.4, Section 8.4, Problem 30

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^2+5)^(3/2)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^2+a^2)^(3/2)du= u/(a^2sqrt(u^2+a^2))+C
By comparing "u^2+a^2 " with "x^2+5 " , we determine the corresponding values as:
u^2=x^2 then u = x and du = dx
a^2 =5 then a = sqrt(5) .
Plug-in the corresponding values on the aforementioned integral formula for rational function with roots, we get:
int 1/(x^2+5)^(3/2)dx =x/(5sqrt(x^2+5))+C