Question

sin z 21) 20) Cosx dx (r? + 1) X 22) sin mx dx 2(x² + a²² (a > 0, b>0) 23) cos ex - cos bx -dx x?

use residue theorem to evaluate the following integrals

Answer 1

use residue Theoren to evaluate DATE The following integral 2 202 CosX dx 2 o (21) 2 solchon Consider the integral 17 2 iz a dea J ce F ( z dz = /4 (cz C (2²71) C contour consiting of Where c is the a semi circle F Radin R targe enough to include -cw the pole integrend plane and also past of the real anis from n=-R to note Show in to digure below in tu Upper batt L R Q ~R R Hence by Cauchy s Rellore Thcopy iz flw dr = 1x findt + S f(z) dza zri ERT S 2 { e I

DATE re ob ud denotes Poles withine Where ERT the residue Plz) Which the lig etc (zn)? whicet Now Take 4 127 e a pole of within C Ordes REJO 2 1 lies by Res (2=0) Obtaine Putton to do thes collecting to coétted Anytz) of ilkt) е Ylital (tti Aftet 02 3 Latt (t²- 2it) 2 el ti e -4f2 ilettº(115 (1-7 Heat-2t et is +317 :) 11t177 -ut t-it T -el -412 = e- 462 [-1-3t) (tit (allt cockbrat a ť i e-1-37a (-1) = e

od s 2 ina a e ooda = 20 i Ent (27) =znice) att i se 00 2 in 2 das T l ze In 2 2 2 cavating real con marion part on both sides da fasina --00 sa x² cdx da = 0 TT ge (141) 2 2242 x2 Cose da = 0 2 (رة) 21) | Sinn da x 0 solution e 1oz -da- 2 C с Where cis Considen the integral I fledz consisting od r to R half of the circle 121ZR w real any h) The upper CR

DATE -R to-r (1) to real any (۱) The upper hall of the circl 12128 Show in figure below X CR R r Zoo CA rR since f(a) has no singularity In sede clits only Singerle point being simple pole at which has been deleted drawing С by Cauley rendre renes R we nare 24 ca -R Stine fins dat J + (2) d2 f I teasent ) 123d2=0 I f(2) d2 - I 16 Now e AR e U i R Ccofot(no) 10 Rie do Re + TJ ir llogorimo i Ricoot mol mo Te ARSIDO ir logo - RSind - Rs 0 do Since 1512/d2 TER D de t 5

ROB 2 2 2 2 0 25" ZRO * do Colo</2 con gyott leh) T which cy R 49 Also I field2 co Petre copot things 4 do ié et og ryo 2 Henc al ro Ryoo we get From o .O o s tindatot troldo -ello to find it ela anzil a cavating real and making pa foar Both side wor Ja sine exit 1 Hono sia a aa 2 ५

DATE 3 Sirona da 22 اما را 0 а (-2, y2 solution consider the integral zlezza f(e) dz elona where f(2)= the given functiva hay poles at 2=0,2=114 0 to real bry besides hast zalia ý pole tooder a) out of the de z = ia cobores two te es withe) Conto use be we a large semi.circle 1ZIER rodenter at z=0 end the thi Soal radicy of circle trpentahon Figure below Choose the f -T -r R in /12) f(2) and the put 2 tia coefficient 그 shall be the recured rende

I of Cees Ti ri enam ? au l'mon or e 00 2a3 002 I sonen dy Xina 22 Ti ema (me auating imaginary on both sedes Joo sim ma ( mez 2 De an 200 pary we set Te the Tau an x (027 72 هل 0 ㅍ T sinma da (mez of real gan 4a3 o a1ntaya

DATE (oo) I todarf flugdrt / fonde f(2) dra fu = f (2) de sino V 7374 +1z) 0 = -21 i Lé (mondes 4.3 Seime 2/27aza ht =0 2(2² a 3 2 ht azo Roa F 2 22a52 132 s Halde s is described inclou director It e f(e) = ema 그 Zo zod Seime white I Q 4 und Lt z(27a²2 Te ni f (2) d2= -itrolan ay we have TO lot of Choose -ve sign In to rew ry described an cloull wise directen Hend making Rooneryo we har from tJt a + 2-17 (e Gaz Ia tindy Sten10470 ma (mp3)