Question

Integrate the function $\frac{1}{\sqrt{x^2+2x+2}}$

Answer 1

$\frac{1}{\sqrt{x^2+2x+2}}dx=\int \frac{1}{\sqrt{(x+1{)}^2+(1{)}^2}}dx$
Let $x+1=t$
$\therefore dx=dt$
$\Rightarrow \int \frac{1}{\sqrt{x^2+2x+2}}dx=\int \frac{1}{\sqrt{t^2+1}}dt$
$=\log |t+\sqrt{t^2+1}|+C$
$=\log |(x+1)+\sqrt{(x+1{)}^2+1}|+C$
$=\log |(x+1)+\sqrt{x^2+2x+2}|+C$