# Don't forget the +C

Calculus Level 3

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

$\color{#D61F06}{\text{Paloma:}}$ This test is simple. Find the integral.

($\color{#D61F06}{\text{Paloma}}$ writes this integral on the whiteboard) $\int\int_0^{\ln N}\dfrac{4e^{2u}(e^{2u}-1)}{9}du\text{ }dx$ $\color{#3D99F6}{\text{Ryan:}}$ 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.

($\color{#3D99F6}{\text{Ryan}}$ writes on his paper) Let $x$ be the answer to this question. $x$

$\color{#D61F06}{\text{Paloma:}}$ Wrong! You forgot the $+C$!

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$\underline{\text{Problem}}$

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

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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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