|I_4|=\left|\int_0^\pi i e^{iaRe^{it}}\frac{1-e^{2iRe^{it}}}{Re^{it}}\,dt\right|\leq \int_0^\pi \left|i e^{iaRe^{it}}\frac{1-e^{2iRe^{it}}}{Re^{it}}\right|\,dt\leq\int_0^\pi\frac{1+|e^{2iRe^{it}}|}{R}\,dt=\int_0^\pi\frac{1+e^{-2R\sin t}}{R}\,dt\leq\int_0^{\pi}\frac{1+1}{R}\,dt=\frac{2}{R}\pi\