# It can't be zero!

Calculus Level 3

Lilly comes to question 2017 on her calculus final:

$\displaystyle \large \int_{- 1}^{1} \sqrt{1 + x^2} \ dx$

She thinks to herself, "This looks like a standard u-substitution question!" and begins to work it out. "Let's see. I'm going to rewrite the integral by letting $u = 1 + x^2$ and changing the limits of integration.

For $x = - 1$, we have $u = 1 + (- 1)^2 = 2$. For $x = 1$, we have $u = 1 + (1)^2 = 2$. So our new integral is..."

$\displaystyle \large \int_{u \ = \ 2}^{u \ = \ 2}$

"Oh wait!" she interjects, stopping herself. "The limits of integration are the same so the integral must equal $0$ no matter what the integrand is!" However, she realizes that $f(x) = \sqrt{1 + x^2} > 0 \ \forall \ x$. "As long as $b > a$, $\displaystyle \int_{a}^{b} \sqrt{1 + x^2} \ dx$ must always be positive. What did I do wrong?"

Can you help Lilly? If the value of the definite integral can be expressed as $\displaystyle \sqrt{m} + \sinh^{- 1} (n)$ for positive, square-free integers $(m, n)$, find $m + n$.

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