How to complete the square of this polynomia

Question

let:

$$\int \frac{dx}{x^2+4x+5}$$

I am trying to complete the square in order to use trig substitution but I failed I was trying:

$$(x^2+4x+1)+5-1$$ $$(x^2+4x+1)+4$$

Answer 1

Given

$$I=\int \frac{dx}{x^2+4x+5}$$ $$\Rightarrow I=\int \frac{dx}{(x+2)^2+1}$$ Let substitute $(x+2)=t$ then we can see

$$I=\int \frac{dt}{t^2+1}$$ $$\Rightarrow I=\arctan(t)+c$$ So $I=\arctan(x+2)+c$ where $c$ is an arbitary integral constant