I need a kickstart with evaluating this integral:
$\int_0^\infty \! \!\int_x^\infty \frac{1}{2}e^{-x-\frac{y}{2}} \, \mathrm{d}y \, \mathrm{d}x$ I can't remember how to solve it, mainly because of $-x$ part.
Thanks in advance.
I need a kickstart with evaluating this integral:
$\int_0^\infty \! \!\int_x^\infty \frac{1}{2}e^{-x-\frac{y}{2}} \, \mathrm{d}y \, \mathrm{d}x$ I can't remember how to solve it, mainly because of $-x$ part.
Thanks in advance.
First off, by linearity we can pull out a factor (recall that $e^{a+b}=e^ae^b$):
$\int_0^\infty \int_x^\infty \frac{1}{2}e^{-x-y/2}dydx= \frac{1}{2}\int_0^\infty e^{-x}\left(\int_x^\infty e^{-y/2}dy\right)dx.$
What's the antiderivative of $e^{-y/2}$ with respect to $y$? Apply the fundamental theorem of calculus...