Note that this was used in the context of string theory so a lot of mathematics is just taken as things appear to be : (Oh..it's -1/12 and if we assume the cessaro mean it works!! OMGZZ STRING FEORY!!)
In a mathematics sense, obviously it's not rigorous at all and is wrong.
Re: MX2 Integration Marathon
\sqrt{\dfrac{1+x}{3+x}} = \sqrt{\dfrac{1+x}{3+x}} \cdot \dfrac{\sqrt{1+x}}{\sqrt{1+x}} = \dfrac{1+x}{\sqrt{(1+x)(3+x)}} = \dfrac{1+x}{\sqrt{x^2+4x+3}} and you should be able to go from there.