Chọn phương án C.

Theo đề bài ta có $$\begin{aligned}
F(x)&=\displaystyle\int F'(x)\mathrm{\,d}x=\displaystyle\int\sin2x\mathrm{\,d}x\\
&=-\dfrac{1}{2}\cos2x+C.
\end{aligned}$$
Vì \(F\left(\dfrac{\pi}{2}\right)=1\) nên $$-\dfrac{1}{2}\cos2\cdot\dfrac{\pi}{2}+C=1\Rightarrow C=\dfrac{1}{2}.$$
Vậy \(F(x)=-\dfrac{1}{2}\cos2x+\dfrac{1}{2}\).