Sometimes, we need a precise value for following

As such, I am going to calculate it and place the result here for future works.

In order to evaluate the above integral, we need to use the so-called co-area formula. We first write

Note that

Therefore,

where we have used

In particular, by the duplication formula, we get that

where is the volume of -sphere of radius 1 in .

Another particular formula occurs when . We then have

where is the volume of -ball of radius 1 in .

Keep in mind that the -sphere of radius 1 is nothing but the boundary the -ball of radius 1. Therefore, is nothing but the surface area of the -ball of radius 1. Interestingly, we have the following well-known formula

As such, the previous formula can be expressed as

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Hi Ngo, how can you do with the same integral above times log(1+|x|^2)? Give me an advice, please!

Txs

Comment by Fab — April 23, 2012 @ 21:21

Unless there is some motivation, otherwise, it is not worth considering that integral. As you may know, the one I considered above comes from blow-up analysis for PDEs.

Comment by Ngô Quốc Anh — April 23, 2012 @ 21:47