Math problem with partial derivatives

This integral took me to solve more than an hour without success :(

In two of your videos you prove i^i = e^(-Pi/2) and the ith root of i, I^(1/i) = e^(Pi/2). When I multiply i^i by i^(1/i) I get i. But when I multiple the representations e^(-Pi/2) and e^(Pi/2) I get e. So, does i = e? Where have I gone wrong?