If we can find the antiderivative function #F(x)# of the integrand #f(x)#, then the definite integral #int_a^b f(x)dx# can be determined by #F(b)-F(a)# provided that #f(x)# is continuous.
We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that #f(x)# is continuous and why.
FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.
For most students, the proof does give any intuition of why this works or is true. But let's look at #s(t)=int_a^b v(t)dt#. We know that integrating the velocity function gives us a position function. So taking #s(b)-s(a)# results in a displacement.