Brilliant integral calculation

But that's the standard way that "real mathematicians" solve easy problems: show the problem is equivalent to a harder problem that has been solved already.

 

The notation seems completely fine to me. What is excessive about it? The parenthesis around the 1/12?

 

I don't think it's excessive. Certainly, multiplying everything out would make the integration dead simple, but his/her approach was pretty concise and interesting. Sometimes, a little extra complexity makes for a much more interesting argument (e.g. my favourite proof of the infinitude of prime numbers uses topology).