$\displaystyle \frac{dz}{dt} + e^{t+z} = 0 $

I moved the $\displaystyle e^{t+z}$ to the right side of the equation and came out with.

$\displaystyle \frac{dz}{e^z} = -e^t dt$

Am I right so far, before I take the integral of both sides?

And can the integral of $\displaystyle \frac{1}{e^z}$ be taken without using parts?