[latex]int frac{y, dy}{sqrt{3y^2+1}}=frac{1}{6}int frac{6y, dy}{sqrt{3y^2+1}}=frac{1}{6}int frac{d(3y^2+1)}{sqrt{3y^2+1}}=\\=[, int frac{dt}{sqrt{t}}=int t^{frac{-1}{2}}dt=frac{t^{frac{1}{2}}}{frac{1}{2}}}=2t^{frac{1}{2}}+C=2sqrt{t}+C, ]=\\=frac{1}{6}cdot 2sqrt{3y^2+1}+C[/latex]