Question 5 please write clearly and explain all the solutions!

I solved question 4 parts a and b. I need you to redo parts b and a! Thank you!

4. (20 points) Use standard integration techniques from elementary calculus to evaluate where the (open) contour y runs from = = -r to = = +r along either a. the portion of the real axis = = r with -r S r S r, or b. the semi-circle = = Firein the lower (upper) half-plane with -*/2 5 0 s #/2, with r > 0 arbitrary and not equal to 1. Hint: In case (b), first use symmetry arguments to show that 2r(1 - 2) cose do (1-72)2 cos2 0 + (1 +r?)> sin* 0' Then, setting r =: tan p, change the integration variable to o such that tano = tan 2psin0. How does the result for I, depend on whether r 2 1? 5. (10 points) Redo part (b) of the previous problem using the result of part (a) and the Cauchy integral formula. 4) Now use standard integration techniques from calculus, 1+27 If where runs from je- to 2+ 8 along either we know by port la) that Iys - [tan '(x]]y - 8 Iy: [- tan - 1(ixe Fie 17 12 along 4: 2- * 30 Iy = Ji ide"=y (icoso Fisina)r/ "TY - [ tan -' ( * ) ] " - ton ()- ton" (- 8 ) ise at - 1= FX Iys - [tom " (F8)- Can '(Fx)] b) 2 = Fixer10 da- five"(ido ) = Fre"do Now, so along the circle: Then, using solution of part (a) izetie "=* , ize"(ide) = dx Now, substituting the values again in integral