As I understand it, N[x] calculates x using machine precision and keeping no track of accumulating errors. Instead, N[x,n] calculates x with all whatever internal precision required to get the result with n correct digits, keeping track of errors. The situation is complicated and you can get apparently contradictory results:

x = ((-5551459608 + 3925475120 Sqrt[2]) Sqrt[2 - Sqrt[2]])^2;
y = 174288;
N[Expand[x]]
N[Expand[x], 6]
N[Expand[x], 7]
y < x < y + 1
y < Expand[x] < y + 1
y < N[Expand[x]] < y + 1
y < N[Expand[x], 6] < y + 1
y < N[Expand[x], 7] < y + 1

Hi Leon,

Please edit your question and add the code so it can be copied/pasted and reproduced.

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int = Integrate[1/(x^3 - 1), x];
Map[Framed, int, Infinity]

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