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`

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