# 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.

$$\color{red}{\text{Paloma:}}$$ 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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