I encountered a definite integral during my research: $$\int^{Q_H}_{Q_L} \text{sinc}^2(aq) e^{2 \pi i \frac{ q\mathbf{x} \cdot (\mathbf{u_1} + \mathbf{u_2})}{N}} dq,$$ where $\mathbf{x},\mathbf{u_1},\mathbf{u_2}$ are 2D-vectors, and $q$ is a real number in the interval $［Q_L,Q_H］$
How can I do this integral? I think it is similar to Fourier transform with finite interval, but I am not sure.