I am trying to evaluate the leading order behaviour of $$I(x) = \int_{0}^{1} e^{ix(t-\sin(t))} dt,$$ using the method of stationary phase. The way we have been taught to solve these types of integrals is find all stationary points in the range, Taylor expand $t-\sin(t)$ about the maxima and minima, and evaluate the integral at small intervals around these points.
I'm running into trouble though because the only stationary point of $$f(t) = t - \sin(t)$$ in the interval $[0,1]$ is $t=0$, which isn't a minimum or a maximum but an inflection point. If anyone could help me out on how to solve this integral I would be very grateful.