Question

3. Choose one of the following integrals, and evaluate it: 21/2 cos x2 – sin x2 dc, 28 +1 x2 +1 0 You are strongly encouraged to also do the other integral for practice! (Hint for the first integral: try evaluating Soo 11 dx.]

THIS IS FROM COMPLEX VARIABLES

USE APPROPRIATE THEOREMS AND INDICATE ALL STEPS CLEARLY. THANK YOU

Answer 1

# SOLUTION!- consider the Integral. T S Ja da x2 +1 ( T 1+z2 u Also Let zb dz where b=2 1+22 and c is the contour having a lorge Semi-Ciride in the upper half plane indented at the arigin. Here f(z) = zb Marly z= ti are simple fall of flz, and only z=1 lies inside c. So, Residue at (z=i) = lim, (z-i, fiz) = lim, zb 2- i zti = 1 / [cushti Sing Now, By Residue therum, we have şfezidz = (A finson + I firsdz + & fcxeingetron R. ib Zti ai ₂ (t) b= 1/2 2521 8 R + Se fizidz ani ERT -(1) Now, If fizidel rb eino ir eld RB +) do poss" 0 1+R2 e 120 R2-1 Rb+) TT-20 R2-1 as R-700 (b=1/2) 29

Similarly Jo ->0 as r >0 (!= 1) Now, from Eun (1) To findn- go froncin, du= ani. Molto lotion) 21 12 poo gb on + 2 eibt ab o 1+x? da = T(1+i) 1+x2ein va Couetu'ny ucl parut on both sides, we get so 20(1+(USTD) on TT 1 +012 Ja so ab da da = T / iz THIUSTb) Ita2 PW D = 1/2 Ve dn = 1/5 (1+1) 1 + x2 = ) ux dn = Answer. 1422 212