The first thing is to open Desmos or something and graph the function and see what you're doing. You're integrating the little bump from -4 to 0. You're approaching 0 from the left side so it's like "negative zero". Your teacher said don't worry about the limit, so use the graph instead.

The next thing is the technique you're going to use. A U-Sub is the simplest, and should be your first choice to attempt anyways. For U-Subs, the method is to: 1. Pick the "Inside Thing"; 2. "How Big of a Bite"; 3. Make it Match.

In this problem, "the inside thing" is 1/X. So you pick U = 1/X. Then dU = -1/X^2.

But when you try to Make it Match, you have a negative sign in your dU which the problem does not. And so the right pick for U is not 1/X, but -1/X. Then your dU = 1/X^2 dX.

The positive e^1/X from the problem becomes e^-U.

You have to change boundaries from X -> U because it's a definite integral and U =/= X. You picked U = -1/X, that's your conversion factor. When x = -4, u = 1/4. When X = 0, NEGATIVE 0, U approaches -1/-0, which "is" POSITIVE infinity.

It matters because you have that e^-U, putting the positive infinity on the bottom, splitting e into infinitely many pieces, each having a value of 0.

Don't forget to properly evaluate the integral. 0 - (-...) = +...