Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 20

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
The format of the given integral problem: int 1/(x^2+4x+8)dx resembles one of the formulas from integration table. Recall we have indefinite integration formula for rational function as:
int 1/(ax^2+bx+c) dx = 2/sqrt(4ac-b^2)arctan((2ax+b)/sqrt(4ac-b^2)) +C
By comparing ax^2 +bx +c with x^2+4x+8 , we determine that a=1 , b=4, and c=8 .
Applying indefinite integration formula for rational function, we get:
int 1/(x^2+4x+8)dx =2/sqrt(4(1)(8)-(4)^2)arctan((2(1)x+(4))/sqrt(4(1)(8)-(4)^2)) +C
=2/sqrt(32-16)arctan((2x+4)/sqrt(32-16)) +C
=2/sqrt(16)arctan((2x+4)/sqrt(16)) +C
=2/4 arctan((2x+4)/4) +C
=2/4 arctan(((2)(x+2))/4) +C
=1/2 arctan((x+2)/2) +C