Question

I= ∫ ( y /(√((2y^2)+1)) )dy

Answer 1

substitute u=2y²+1 and du= 4ydy:

$$\frac{1}{4}\int \frac{1}{\sqrt{u}} du = \frac{1}{4} 2\sqrt{u}+const=\frac{\sqrt{u}}{2}+const$$

So

$$\int \frac{y}{\sqrt{2y^2 +1}} dy=\frac{1}{2}\sqrt{2y^2 +1}+const$$