1. Disguised Bernoulli.

I'm sure this is a very basic differential equation but I cant figure out how to separate it to begin with.

$\frac{dy}{dx} = e^{x + y} + e^x\sin{x}$

2. Note that $e^{x+y}=e^{x}e^{y}.$ Then substitute $z=e^{y}.$ I think you'll find that the resulting DE is Bernoulli.

3. Thanks, thought it was that.

4. You're welcome. Incidentally, I don't think you'll be able to find a closed-form solution of this DE. That is, you'll still have integrals in your final solution, because there are no known antiderivatives for the integrals you get. That's ok. You still get an explicit solution (all the x's on one side), which is nice.

5. Oh sorry I mistyped the problem in the OP, it's actually:
$\frac{dy}{dx} = e^{x + y} + e^y\sin{x}$
from which I get...

$\int \frac{dy}{e^{2y}} = \int e^x + sinx dx$

$\frac{e^{-2y}}{2} = e^x - cosx$
and finally...

$2y = \ln{\cos{x}} - x$

Is that correct?

6. The two sign errors that you made evidently corrected themselves in your final answer. Don't forget the constant, too.

[EDIT]: Also, I don't think you're handling the factor of 2 correctly.