It can't be zero!

Calculus Level 3

Lilly comes to question 2017 on her calculus final:

111+x2 dx\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+x2u = 1 + x^2 and changing the limits of integration.

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

u = 2u = 2\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 00 no matter what the integrand is!" However, she realizes that f(x)=1+x2>0  xf(x) = \sqrt{1 + x^2} > 0 \ \forall \ x. "As long as b>ab > a, ab1+x2 dx\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 m+sinh1(n)\displaystyle \sqrt{m} + \sinh^{- 1} (n) for positive, square-free integers (m,n)(m, n), find m+nm + n.

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