$\int \sqrt{x^2+2x+5}\,dx$ equals

1. Complete the square: $x^2+2x+5=(x+1)^2+4$. 2. Put $y=x+1$: integral $=\int \sqrt{y^2+2^2}\,dy$. 3. Use $\int \sqrt{y^2+a^2}\,dy=\dfrac{y}{2}\sqrt{y^2+a^2}+\dfrac{a^2}{2}\log|y+\sqrt{y^2+a^2}|$ with $a=2$. 4. Back-substitute: gives option A. Option B wrongly uses $\sin^{-1}$ (that is the $a^2-x^2$ form). _Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.331_