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residue calculus evaluation of infinite integrals

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iR$IZ ]sj{ZjZR e%sjZs$Bt BP ytht$ ytsjR ytsjR BP ]s$t is$Btsj 3Zt{$BtR G {BtR$I s$Btsj PZt{T $BtR BP s {B8}j- %s$sAj F s,$t Y PB8 WwFD) wFD RwFD , bY wFDstIRwFD s }BjktB8$sjR $t F b$Y sj {B^{$tRU wFD) F L  F L NNNL  F L  ,E (, RwFD)R F L R F L NNNL R F L R ,z E L13 yt sII$$Bt b RZ}}BR Ys RwFD YsR tB 9BR Bt Y sj s-$R G bst B %sjZs $tsjR BP Y PB8 8 Ww?D Q? ) 8 w?D Rw?D Q?3 7$t{ RwFD YsR sj {B^{$tR stI YsR tB 9BR Bt Y sj s-$R sjj BP $R 9BR B{{Z $t {B8}j- {Bt`Zs }s$R stI b {Bt{jZI z )1"G8sk RZ}}BR Ys RwFD YsR 9BR F , F ,F, F , NNN,F , F , stI b b$jj RZ}}BR Ys F ,F, NNNF s Y 9BR $t Y Z}} YsjP }jst G {BtRZ{ s }BR$$%jk B$tI X}sY B X{BtBZ" a {BtR$R$t BP Y $t%sj >, H Bt Y sj s-$R BY b$Y Y R8${${j ? ) \F //F/ ) , i Fk( i3 d 3ZY b R${ {BtR$Is$Bt B %sjZR BP  RZ{Y Ys k/F /,>) d, 1, 333, " V}}jk$t Y iR$IZ qYB8 b Ys% 8 wFD RwFD QF )1$7 C H 3 iR wFD RwFD e e e e e I O # 1$7 3 ' 3 qYt 8 w?D Rw?D Q? )1$7 3 '  8 wFD RwFD QF3 qY $Is tBb $R B RYBb Ys j$8 $ QF ) ( qB Y$R tI b BAR% Ys e e e e e wFD RwFD e e e e e ) d /F/ e e e e e  L  F L NNN L  F L  F R L R F L NNN L R F L R F e e e e e 3 PB8 bY${Y $ $R {js Ys Y s }BR$$% tZ8AR  stI  RZ{Y Ys 8s- e e e e e wFD RwFD e e e e e    ,  3 qYt b {st ZR Y R$8s %sj$I PB     e e e e e 8 wFD RwFD QF e e e e e  8s- e e e e e wFD RwFD e e e e e jtY w? D    $ ) $  B R Ys $ $R $tII Z Ys j$8 $ QF ) ( qYs A$t Y {sR b tBb Ys% 8 w?D Rw?D Q? ) j$8 8 wFD RwFD QF )1$7 3 ' , Y ' ,> )d, 1, NNN," A$t Y R$IZR BP jk$t $t Y Z}} YsjP }jst G {YBBR sR BZ hR -s8}j st $tsj bY${Y {st A %sjZsI ZR$t Y RstIsI 8YBIR BP {sj{ZjZR 1 e-s8}j d G %sjZs Y $tsj $  3B8 Y RstIsI {sj{ZT jZR b Ys% 8 Q? dL? ) j$8 st w?D e e e e ) $ 1  w  $ 1 W ) $3 =R$t Y R$IZ {sj{ZjZR b tB Ys ) YsR s R$tj $RBjsI R$tZjs$k $t Y Z}} YsjP }jst s F ) 7 qYs R$tZjs$k $R s }Bj BP BI d RB Y R$IZ Y $R j$8 wF 7D d wF 7DwF L 7D ) j$8 d F L 7 ) d 17 3 qYt 8 Q? dL? )1$7 d 17 ) $3 qY t- -s8}j $R Bt PB bY${Y Y ZR BP Y RstIsI {sj{ZjZR bBZjI A RZARst$sjjk 8B I$^{Zj e-s8}j 1 G {B8}Z $  qY 9BR BP R wFD)dLF s Y PBZY BBR BP d !P YR Btjk F ) dL7 $ 1 ,F ) dL7 $ 1 , j$ $t Y Z}} YsjP }jst G$$t F dLF ) F wF F DwF F D p F L $ 17Fd Q , Y R$IZ s F ) $R j$8 wF F D F wF  F DwF  F D p F L $ 17F d Q ) j$8 F wF  F D p F L $ 17Fd Q n ) 7 p  Qp 7 L $ 17  d Q ) 7 1 $ 1w7  dD ) d  7 ; $ 1 3 V R$8$js {B8}Zs$Bt $%R Y R$IZ s F ) sR )  qY %sjZ BP Y $tsj $R YZR 8 ? Q? dL? )1$7 X d  7 ; $ 1 L d  7 ; $ 1 ~ ) $ $ 1 3 e%sjZs$Bt BP ytsjR BP Y 3B8 $  w?D Rw?D l {BR ? R$t ? M Q? < b 8s, Y Rs8 sRRZ8}$BtR Bt w?DstIRw?DsR$tYPBBT $t I$R{ZRR$Bt AZ tBb $ $R tBZY B sRRZ8 z  E L d Y sjts$t {Yss{ BP Y $BtB8${ {B8}Btt BP Y $tstI sRRZR Y -T $Rt{ BP Y $tsj sR jBt sR $R Zj$8sjk 8BtBBt I{sR$t B 8BtBBt $t{sR$t B (u stI Ys $R Y {sR ZtI YR {${Z8Rst{R VRRZ8$t k(bZRYRs8{BtBZa sR }%$BZRjk w$P c( b ZR s {B8}ssAj {BtBZ $t Y jBb YsjP }jst $P  ) ( b Zt B Y sj$ I$R{ZRR$Bt b$Y z  E L 1D G 8$tI BZRj%R Ys 8 wFD  QF RwFD ) 8 wFD{BRF QF RwFD L 7 8 wFDR$tF QF RwFD stI YZR 8 wFD{BRF QF RwFD )i X 8 wFD  QF RwFD ~ , 8 wFDR$tF QF RwFD )y8 X 8 wFD  QF RwFD ~ 3 yt Y$R R$Zs$Bt sY Yst {BtR$I$t `ZR Y R8${${j ? sR sj$ b I{B8}BR Y {BtBZ a $tB Y $t%sj >H sjBt Y sj s-$R Y ; R w  {BtR$R$t BP bB RZATs{R BP ? IhtI Ak w ) \F ) ? L 7r ;?/ ( cr  /?/ k stI F ) \F ) ? L 7r ;?/ ( c  /?/ cr k 3 yt Y Z}} YsjP }jst rk( b Ys% e e e  e e e ) e e e   e e e )  , bY${Y $t }s${Zjs $R jRR Yst d =R$t 8Z{Y Y Rs8 R$8s sR $t Y sj$ I$R{ZRR$Bt b {st Yt R Ys PB    stI & {YBRt RB Ys Y jtY BP Bt BP Y s{R BP w $R jRR Yst B JZsj B wd L &D $  PB     e e e e e 8 wFD  RwFD QF e e e e e  8s- e e e e e wFD RwFD e e e e e jtY ww D  1wd L &D   $  )1wdL&D  $  3 y $R {js Yt Ys j$8 e e e e e 8 wFD  RwFD QF e e e e e )(3 !t F  Bt Y BY YstI b Ys% e e e  e e e ) e e e   e e e )    , stI b R Ys e e e e e 8 wFD  RwFD QF e e e e e  8s- e e e e e wFD  RwFD e e e e e jtY wF D     (, 3 ]B8A$t$t Y bB RZjR b R Ys j$8 8 wFD  RwFD QF )( x RB Ys s,$t Y j$8$ sR   `ZR sR APB stI Iht$t Y R$IZR ' sR APB 8 w?D  Rw?D Q? )1$7 3 ' 3 qYs A$t Y {sR b Yt Ys% 8 wFD{BRF QF RwFD )i X 1$7 3 ' ~ , 8 wFD{BRF QF RwFD )y8 X 1$7 3 ' ~ 3 e-s8}j n G {B8}Z $  ? R$t 1?Q? dL?  qY {BR}BtI$t {BtBZ $tT sj $R $ a F QF dLF  qY Btjk R$tZjs$k BP Y $tstI $t Y $t$B BP a PBkd B{{ZR s F ) 7 bY Y R$IZ $R j$8 wF 7D F wF  7DwF L 7D ) j$8 F F L 7 ) 7 17 )  1 3 qYZR 8 ? R$t 1?Q? dL? )y8 X 1$7  1 ~ ) $ 3 ]BR}BtI$tjk 8 ? {BR 1?Q? dL? )i X 1$7  1 ~ )(, bY${Y $R {js $t stk {sR A{sZR Y $tstI $R st BII PZt{$Bt BP ? E

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