Question

Can anyone help me integrate e^-(x+y) for x,y>0

Answer 1

\[\int\limits_{?}^{?}e^udu = e^-(x+y) + C\]\[\int\limits_{?}^{?}\int\limits_{?}^{?}e^(-x-y)dxdy\], let u = -x-y, and take the partial derivative wrt x so du = -1\[\int\limits_{?}^{?}-\int\limits_{?}^{?}e^ududy\] \[\int\limits-e^(-x-y)dy\]. Now let u = -x-y and take the partial wrt y, du=-1 so \[\int\limits_{?}^{?}e^udu=e^-(x+y) + C\]

Answer 2

you can do either dy or dx first, but it should still come out the same. Just remember that the u sub eliminates the second variable in du, because the partial derivs treat the other variable as a constant. Kind of like subbing u = 2x+3, du = 2dx, but in this here it's with x and y

Answer 3

I think I get it. That subbing makes me dizzy but it'll catch on. Thanks.

Answer 4

just remember, when subbing with two or more variables, your du's consist of partial derivatives.

Answer 5

Thx