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R R 5. To compute 1 = lim 2 COS dr and J = lim 22+1 sinc dx simultaneously .22 +1 R R0 R R using Residue Theorem, let f(x) 22
C COSC sinc (1) Show that if z = x + iy, then Rf(R2) = and Sf(R2) = x2 +1 x2 +1 (2) Find Res[f, i]. (3) Show that I = 0 and J
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Solution R Given ; doc I= lim Ryoo x Cosx x² + 1 R Labo dx ac Sink Lim Rtoo xc² + 1 -R zeiz 1) Take f(z) : z2 + 1 vo where, IAnd, Res (f(2), i) = Lim (2-2) f (2) zyi alim (2-1) zeiz zi z²+1 ze zeiz zti Zti i ciri iti elo or Res (&(2), i) = a ૨ ૯ 3) AR Given, x Cosse doc H 11 lim R700 x² + 1 -R take f (x) = ac Cosse 20²+1 So, f(-x) = (-x) Cos (-x) (-x)² + 1 x Cos (x) x² + 1

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