Question

Problem 3. Evaluate the integral co sinx dx. Hint: Apply residue theorem to the function f(z) = and the contour y of the following shape:

Answer 1

Given the on sinn an Solution: we know that sina is the imaginary part of ein Therefore, consider the function f(z)= e2 now, fie) has a simple pole at zo lies This pole will be avoided by indentation on the real anis. Draw a small semi-circk er: 121=8 with contours the Singular point zzo inside a semi-circle cri RAR. - R - Y Now, evaluate the integral of reap 2, where a consists of anis rto R, the semi-circle ce, real anis ar to - parts Small semi-cirde ar, since c is the closed contour and a singularity, we have, by canchy's theorem and has no Stredz=0 fedz 20 je festa di tes30 + Frete af forsmes consider sommes pre es Bu zamansaray - | cat simo) e como los senia) R (col@+13140) do Zekelo dz=Riello

| 492s}"easing 2)dz TT RSD do I e Rsino Odo Note that when o increases from oto I sine decreases from 1 to 7 Ft e for asosh sing >, 2 tr or sinos, 20 =(16**]* F(-a) which tends to o as R70. do Also & fc3d2= 1 fer costising) which tends to if de = it as 8-) Thus, when R-700 and when 8-) from (1) , we have . . . Lºw) - + J )++ cm) =e - I finsona itt jo ell onzint =) se casa tuşunu duzin Equating the imaginary parts of both sides weger o sinn an at