Interchange the order of integration and evaluate the integral [0.1]∫ [x,1]∫𝑒^𝑥/𝑦 𝑑𝑦𝑑𝑥

Answer 1

∫[0,1]∫[x,1] e^x/y dy dx
= ∫[0,1] e^x lny [x,1] dx
= ∫[0,1] e^x lnx dx
= -1.3179
Note that this integral cannot be done using elementary functions.
To change the order, note that the region involved is a triangle in the x-y plane, so we can write it instead as
∫[0,1]∫[y,0] e^x/y dx dy
= ∫[0,1] e^x/y [y,0] dy
= ∫[0,1] (1-e^y)/y dy
= -1.3179