Question

Evaluate the integral ∫_(-1)^1▒1/√(x^2+2x+2) dx

Answer 1

$\int_{-1}^1 \frac{dx}{\sqrt{x^2+2x+2}}=\int_{-1}^1 \frac{dx}{\sqrt{(x+1)^2+1}} =\\ =[x+1=t, dx=dt, x=-1\implies t=0; x=1 \implies t=2] =\\ =\int_0^2 \frac{dt}{\sqrt{t^2+1}}=(\ln |t+\sqrt{t^2+1}|)|^2_0= \\ =\ln|2+\sqrt{2^2+1}|-\ln|0+\sqrt{0^2+1}|=\ln(2+\sqrt{5})$