Why would one want to partition an integral

Is it because answers are more accurate? For example, what is the difference between the integral from:

0 to 3 for x^2

and the integral: 0 to 1, 1 to 2, and 2 to 3.

They both equal nine so I am wondering why would one want to partition an integral

If you want to find the actual area under a curve you will need to partition because any area below the x-axis will be negative.

In your example's case, there isn't much advantage to breaking up the integral like that. However, if you have something like
∫ (lnx - 1)/x dx, if you were to partition it by separating the integrand into two rational ones, you could immediately evaluate one of them (∫ 1/x dx) then evaluate the other by u-substitution.

Your integral is simple enough to evaluate on its own. However, if the problem in question is an application of physics or something, then partitioning it could get you results at certain intervals.

partitioning is needed for integrating modulus functions. for example integrate |x| from -3 to 3. Without partitioning we cannot get the correct answer.

For this problem, we must partition when x=-3 to x=0 and when x=0 to x=3. In the former x is negative, but |x| is positive, there fore we must integrate -x instead of x in that range. For the later interval, x is positive so |x|=x for that region, ans we should integrate x in that range. For the final ans we should add up the two integrals. without partitioning, it would be hard to do these sort of problems.

To get intermediate results.
For instance, suppose that you have a simulation running. If you integrate and get the final result right away, you will no see the evolution, the progressive change as it plays out.