Question:

Put $x^2 = t \Rightarrow 2x \, dx = dt$<br/>$\therefore \int \frac{2x}{\sqrt{x^4 + 2x^2 + 4}} dx = \int \frac{dt}{\sqrt{t^2 + 2t + 4}} = \int \frac{dt}{\sqrt{(t+1)^2 + (\sqrt{3})^2}}$<br/>$= \ln \left| (t+1) + \sqrt{(t+1)^2 + (\sqrt{3})^2} \right| + C = \ln \left| (x^2 + 1) + \sqrt{x^4 + 2x^2 + 4} \right| + C$