How to calculate this iterated integral:$\int_0^1dy\int_0^y dx\int_0^x \frac{e^z}{1-z}dz$?
Is there any trivial way to calculate this iterated integral,$\int_0^1dy\int_0^ydx\int_0^x\frac{e^z}{1-z}dz$? I've already solved this by taking series expansion of $e^x$ into the integral,but I wonder if there is any other easier way.
Solution 1:
your joint support is
$$0<z<x<y<1$$
thus first integrate in $dy$ obtaining
$$\frac{e^z}{1-z}\int_x^1 dy=\frac{e^z}{1-z}(1-x)$$
now your support is
$$0<z<x<1$$
and you can integrate
$$\int_0^1\frac{e^z}{1-z}\left[\int_z^1(1-x)dx\right]dz=\frac{1}{2}\int_0^1(1-z)e^z dz=\frac{e-2}{2}$$