Doubt from Definite Intergal

#1

My doubt is from step 2 to 3 why in 14+ [x]; [x] isn’t removed as the limits are from 0 to {x} where the value of [x] is zero. As {x} always less than zero.

#2

As the LHS also contains the greatest integer function they thought to maintain the term in RHS also till last of the problem, such that we might not miss some solutions.

#3

I thought that too, cos at the end if we would have removed that RHS GIF, we couldn’t get the solution. My question is just for the sake of getting the answer they might have done it or can we actually remove GIF like I mentioned in some other solution to get the answer.

#4

We can remove GIF, if it's behave like constant in the range, i.e, 0 to {x } which the value is 0 . Just for the sake of the answer they continued it