Well, You have 2 answers already. Butsince u have asked me and since the above solutions appear to be clumsy, let me try my bit.
The give f(t) = (t-t/2)
2 or (t-1/2t)
2
(a) f(t) = (t-t/2)2 = [t(1-1/2)]2 = (t/2)2 =t2/4
Integrating f'(t) =t3/12
Evaluating for t = 1 to 2
f'(t) = t3/12 becomes (8-1)/12 = 7/12
(b) f(t) = (t-/2t)2 = t2 + 1/4t2 - 2tx1/2t
= t2 + 1/4t2 - 1
Integrating, f'(t) = t3/3 -1/4t -t
Evaluating from t= 1 to t = 2
the above integral becomes,
=(8/3 -1/8 -2) - (1/3 -1/4 -1)
= 7/3 +1/8 -1 = 35/24 = 1.46
Pl check the simplifications.