the given vector and the bounds of integration are;

$\displaystyle r_{0}=\langle\sqrt{2},e^t,e^{-t}\rangle; 0\leq t \leq 1$

differentiation gives;

$\displaystyle r^\prime_0=\langle 0, e^t, -e^{-t}\rangle$

definition of the length of a curve (were $\displaystyle | r^\prime_0|$ refers to the magnitude of the vector $\displaystyle r^\prime_0$);

$\displaystyle L=\int_{a}^{b} | r^\prime_0|dt$

substituting the magnitude of $\displaystyle r^\prime_0$ and the bounds of integration into the equation;

$\displaystyle L=\int_{0}^{1} [ 0+ (e^t)^2+ (-e^{-t})^2]^{\frac{1}{2}}dt$

So every problem, of this general form, so far has had a very simple integration where there is ultimately just a constant under the radical. So I think I'm looking for a way to combine the exponential functions into a constant and I can't come up with it so far.

If this is the wrong strategy please let me know.