Two methods to integrate? Are both methods to solve this equation correct? 
$$\int \frac{x}{\sqrt{1 + 2x^2}}  dx$$
Method One: 
$$u=2x^2$$
$$\frac{1}{4}\int \frac{1}{\sqrt{1^2 + \sqrt{u^2}}}  du$$
$$\frac{1}{4}log(\sqrt{u}+\sqrt{{u} +1})+C$$
$$\frac{1}{4}log(\sqrt{2}x+\sqrt{2x^2+1})+C$$
Method Two
$$u=1+2x^2$$
$$\frac{1}{4}\int\frac{du}{\sqrt{u}}$$
$$\frac{1}{2}\sqrt{u}+C$$
$$\frac{1}{2}\sqrt{1+2x^2}+C$$
I am confused why I get two different answers.

Your first method is not correct. I suspect you are trying to use the following result which is not correct in general: $$\int\frac{1}{f(x)}dx = \log f(x)+C.$$
The correct form would be $$\int\frac{f'(x)}{f(x)}dx = \log f(x)+C,$$ which you cannot use in your example because the denominator of your integrand is not a linear function of $x$.
