The question says to use integration by parts on this question, but at the minute we only have one variable.

Therefore, we introduce a 1, so that log|x|= 1*log|x|, here we have not altered the value of the function, but have intoduced a variable so that integration by parts can be used.

The derivative of Log|x| is simply 1/x, so it will be the 1 that we will integrate, which is x.

We then sub these into the by parts formula of uv-∫u'v

This is therefore equal to xlog|x|-∫x/x.dx

=xlog|x|-∫1dx

=xlog|x|-x.