Question

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

Answer 1

$∫\frac {1}{\sqrt {x^2+2x+2 }}\ dx$ = $∫\frac {1}{\sqrt {(x+1)^2+(1)^2} }\ dx$

$Let \ x+1 = t$

$∴ dx = dt$

$⇒ ∫\frac {1}{\sqrt {x^2+2x+2}} dx = ∫\frac {1}{\sqrt {t^2+1}} dt$

$=log\ |t+\sqrt {t^2+1}|+C$

$=log\ |(x+1)+\sqrt {(x+1)^2+1}|+C$

$=log\ |(x+1)+\sqrt {x^2+2x+2}|+C$