Ok, well firstly the acceleration due to gravity is only

on the surface of the moon. The easiest way to think about this is, consider the Earth. Is g constant? No. It varies with altitude,location etc. This is why I put in that statement about the acceleration, that allows you to determine the acceleration for the body at all positions not on the surface of the moon i.e. some distance. They just include that stuff so you can find the constant term. You also have to be careful that the acceleration is always negative because of how you have defined distance with respect to the centre of the Moon's mass or Moon.

To answer your next point, when I said Earth I meant moon, that was a complete fk up

Ok and for part (ii) I skipped a few steps because I was afraid my equation would be too long to pop it in one post. Anyways, all I did was get the equation I derived in (i) but without putting in the limits (think of the integral in (i) as indefinite not definite). Then this is where the logic comes into play. Escape velocity is the minimum velocity required for the body to JUST escape the gravitational force of the Moon and enter space. Hence you assign the condition

, so you will be able to find the minimum velocity for the body to escape! Also technically you should specify direction i.e. away from the Moon's centre of mass etc. So the answer they've given is technically not correct it should be > not equal to it, because when it is equal it will orbit, but that's bordering into the realm of physics.

Here, let me post a picture of my diagram and also of my working.

mechanic q diagram attempt 2.JPG
Hope this helps

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