Paloma gave her Calculus class a very short math test. This is the dialogue between Paloma and one of her students.

\(\color{red}{\text{Paloma:}}\) This test is simple. Find the integral.

(\(\color{red}{\text{Paloma}}\) writes this integral on the whiteboard) \[\int\int_0^{\ln N}\dfrac{4e^{2u}(e^{2u}-1)}{9}du\text{ }dx\] \(\color{blue}{\text{Ryan:}}\) I have no idea how to do this. I can't even come up with a reasonable guess. Maybe I can be smart and get \(\textit{some}\) credit.

(\(\color{blue}{\text{Ryan}}\) writes on his paper) Let \(x\) be the answer to this question. \(x\)

\(\color{red}{\text{Paloma:}}\) Wrong! You forgot the \(+C\)!

\[\text{......................................................................}\]

\(\underline{\text{Problem}}\)

What is the \(N\) Paloma wrote in the upper bound that made the answer to the question \(x+C\)?

\[\text{......................................................................}\]

Author's note: I know that double integrals are supposed to be definite and have functions in two variables. However, coming up with math jokes is hard.

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