\lim_{z\to i}(z-i)f(z)=\lim_{z\to i}(z-i)\frac{e^{itz} }{ z^2+1}=\lim_{z\to i}(z-i)\frac{e^{itz} }{ (z-i)(z+i)}=\lim_{z\to i}\frac{e^{itz} }{ z+i}=\frac{e^{-t}}{2i}.
\int_C f(z)\,dz=2\pi i \operatorname{Res}_{z=i}f(z)=2\pi i\frac{e^{-t} }{ 2i}=\pi e^{-t}.