Another contour integration problem

[outhsakotad]

Homework Statement

See the attached PDF for problem and attempted solution.

Homework Equations

The Attempt at a Solution

According to Mathematica, the answer should be pi, but I keep getting pi/2, and I'm not sure where I'm making a mistake. Hints would be greatly appreciated. Thanks!

 

[vela]

The residue at z=1 should be positive. You need to multiply by (z-1) not (1-z).