I am trying to evaluate this integral $$\int_0^\pi{te^{ix\cos(t)}dt},\ x\to\infty$$ I am asked to use the method of steepest descent to get the asymptotic up to the $x^{-3}$ term.
I get stuck quite at the beginning as i try to find the contour along which i should integrate. I also considered using the stationary phase method but i'm not sure how to get the approximation up to the asked order.
Thanks in advance