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How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t?

$$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$

I know the answer is $$ e^x\sqrt{1+e^{2x}} + e^{-x}\sqrt{1+e^{-2x}} $$ but I'm not entirely sure how to get there. I know it involves using FTC part two to get F(b)-F(a), and I can see you plug in b and a right into the equation, but why doesn't it look like it's an antiderivative?

EDIT: forgot to change the dummy variables

Bosque
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    The dummy variable should not match the variable on the limits. – Clarinetist Feb 04 '16 at 23:58
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    http://math.stackexchange.com/questions/139183/differentiating-definite-integral . Same kind of question; just different integrand and limits. – randomgirl Feb 04 '16 at 23:58
  • Voted to close as a duplicate, but in case the fine points of your question aren't addressed elsewhere: 1. The integral can be "differentiated with respect to $x$", but not with respect to the dummy variable $t$; 2. The derivative of the integral isn't an antiderivative of the integrand, so there's no reason it should look like one. :) – Andrew D. Hwang Feb 05 '16 at 00:55
  • Since we know $f(+t)=f(-t)$, then the problem is reduced:

    $$2\int_0^{e^x}\sqrt{1+t^2}dt$$

     – Simply Beautiful Art Feb 05 '16 at 01:08

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Hint: change the lower bound to 1/x and the upper bound to x, and write the integral as a function of x. Plug e^x into this function and differentiate using the chain rule.

Vik78
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