Let suppose to 'attack' the slightly different integral...

(1)

At this scope we can compute the complex integral...

(2)

... along the red path indicated in figure...

... when and . Since no singularities lie inside the path is...

(3)

... and for the Jordan's lemma the integral along the 'big half circle' tends to if . Tacking into account that the (3) becomes...

(4)

Now if and , tacking into account that ...

(5)

... from (4) we derive that...

(6)

From (6) is evident that is...

(7)

... and, because the integral in first member diverges, the integral in second member diverges too...

Kind regards