It comes from practice with solving good quality problems. After a while you will know what clues to look for.
In the problem solved by Jagdish Sir, he began with exploring by applying the well known property of definite integrals. Because of his experience he knows for sure that it will have some relation to f(x). Thats how he knew what to look for
In the one solved by me, the options suggested that the way forward would involve x=t^3 (the upper limit being 8 confirms the hunch). Then it is also clear that they have split it into two parts. The choice of 1 as the splitting point is then obvious.
In that sense problem solving is far easier than math research where such clues arent that easily available.