It looks to me as though you have spotted an error in this question. The "known" function F(x) only provides information about f(t) in the interval \(\displaystyle |t|\leqslant \pi/2\). So you know nothing about f(x) for \(\displaystyle x>\pi/2\) and therefore there is no way that you can describe its integral in the interval \(\displaystyle [\pi/2,\pi]\).

It looks to me as though you have spotted an error in this question. The "known" function F(x) only provides information about f(t) in the interval \(\displaystyle |t|\leqslant \pi/2\). So you know nothing about f(x) for \(\displaystyle x>\pi/2\) and therefore there is no way that you can describe its integral in the interval \(\displaystyle [\pi/2,\pi]\).