I don't get the first integral. How do they get to x(sqrt(x^2 + ^2 +1)? I would think you would need a ln.
The 0.5 gets cancelled out when you divide by the new power of 1/2.
$\displaystyle
\int \frac{xy}{\sqrt{x^2+y^2+1}}dy$
u = $\displaystyle x^2+y^2+1$
du = 2ydy
$\displaystyle
\int \frac {1}{2}x \frac {2ydy}{\sqrt{u}}$
$\displaystyle =\int \frac{1}{2}xu^{-1/2}du$
$\displaystyle =\frac {1}{2}x2u^{1/2} $
$\displaystyle x\sqrt{x^2+y^2+1}$