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
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.