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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