Sustitución: $u = \sqrt{x - 2} \implies x = u^2 + 2$, $dx = 2u \, du$.
Integral:
$$ \int \frac{2u \, du}{(u^2 + 2)u} = 2 \int \frac{du}{u^2 + 2} $$
Usando $\int \frac{du}{u^2 + a^2}$ con $a = \sqrt{2}$:
$$ 2 \cdot \frac{1}{\sqrt{2}} \arctan\left(\frac{u}{\sqrt{2}}\right) + C = \sqrt{2} \arctan\left(\frac{\sqrt{x - 2}}{\sqrt{2}}\right) + C $$
$$ \boxed{\sqrt{2} \arctan\left(\sqrt{\frac{x - 2}{2}}\right) + C} $$