Integrate |0->1 |e^y->e (x/lnx)dxdy?

Answer 1

Reverse the order of integration.

Draw a picture of the region of integration. It is bounded by the part of the x-axis between x=1 and x=e; a vertical line segment from (e,0) to (e,1); and the curve x=e^y (which is the same as y = ln x). It looks a bit like a triangle, with a curved "hypotenuse". To do y-integration first, this region is described by 0 < y < ln x, 1 < x < e.

Can you take it from there?