Question Number 219574 by SdC355 last updated on 29/Apr/25
$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{sin}\left({z}\right)}{{z}\left({z}^{\mathrm{2}} +\mathrm{4}\right)}\:\mathrm{d}{z}=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\:\infty} \:\:\left(\frac{\mathrm{sin}\left({z}\right)}{{z}}−\frac{\mathrm{sin}\left({z}\right)}{\mathrm{2}{z}+\mathrm{4}\boldsymbol{{i}}}−\frac{\mathrm{sin}\left({z}\right)}{\mathrm{2}{z}−\mathrm{4}\boldsymbol{{i}}}\right)\:\mathrm{d}{z} \\ $$$$\mathrm{and}\:\mathrm{next}….??? \\ $$$$\mathrm{2}\pi\boldsymbol{{i}}\underset{{j}=\mathrm{1}} {\overset{{M}} {\sum}}\:\:\mathrm{Res}_{{h}={a}_{{j}} } \left\{{f}\left({h}\right)\right\}…. \\ $$