Does this integral converge
$$\lim_{R \to \infty}\int_{\frac{\pi}{2}}^{\pi}\frac{e^{Rte^{i\theta}}}{\sqrt{Re^{i\theta}+1}}\cdot iRe^{i\theta}d\theta$$
where t is a positive integer?
If the integral diverges, how can I prove this?
p.s. This integral is part of a larger contour integral to calculate
$$\int_{a-i\infty}^{a+i\infty} \frac{e^{zt}}{\sqrt{1+z}} dz$$
I know that
$$\lim_{R \to \infty}\int_{\frac{\pi}{2}}^{\pi}\frac{e^{Rte^{i\theta}}}{\sqrt{Re^{i\theta}+1}}\cdot iRe^{i\theta}d\theta + \int_{-\pi}^{\frac{-\pi}{2}}\frac{e^{Rte^{i\theta}}}{\sqrt{Re^{i\theta}+1}}\cdot iRe^{i\theta}d\theta=0$$