Double integrals, trying to be sure I am not doing something wrong

[Emspak]

1. evaluate the following:

$\int^{1}_{0}$$\int$$^{1}_{0}$xye^x+y dydx

The Attempt at a Solution

OK, so this should be pretty simple. But for some reason I am having trouble integrating the ye^x+y bit with respect to y. If I do it by parts I end up with iterations like this:

$\int^{1}_{0}$xye^x+y - $\int^{}_{}$e^x+yxdy dx

And integrating again I got

$\int^{1}_{0}$xye^x+y - e^x+yx $|^{1}_{0}$ dx plugging in the relevant values I got

$\int^{1}_{0}$xe^x+1-xe^x+1 - xe^x + xe^xdx

which with the latter terms cancelling gets you $\int^{1}_{0}$xe^x+1-xe^x+1 dxAnd the whole thing should go to zero.

OK, did I do this whole bit correctly? I want to make sure that I am not doing something stupid. If I did it right, wonderful. (I know this might sound kind of silly to post a problem that I think I did correctly, but it never hurts to have another pair of eyes)

 

[LCKurtz]

I don't have time to run through your work right now, but surely something is wrong since your integrand is positive. You must have a positive result. Why don't you try writing $e^{x+y} = e^xe^y$ and separate the integrals. Really easy then.

 

[hsetennis]

1. evaluate the following:

$\int^{1}_{0}$$\int$$^{1}_{0}$xye^x+y dydx

The Attempt at a Solution

OK, so this should be pretty simple. But for some reason I am having trouble integrating the ye^x+y bit with respect to y. If I do it by parts I end up with iterations like this:

$\int^{1}_{0}$xye^x+y - $\int^{}_{}$e^x+yxdy dx

And integrating again I got

$\int^{1}_{0}$xye^x+y - e^x+yx $|^{1}_{0}$ dx

You're correct up to here. After here is where your solution diverges from correct solution.

P.S. It may be helpful in the future to check computationally:

 

[Emspak]

Thanks, I did it again and I realized I forgot the y component. D'OH!