No integration was done. For instance, for the imaginary term, you multiply by the right-hand side by ##\exp[i k_0(x-x_0)] \exp[-i k_0(x-x_0)]##, and move the second exponential inside the integral sign.
No integration has been done yet. It's the old mathematical trick of multiplying and dividing by the same thing.
Outside the integral is a factor of
and inside the integral is a factor of the reciprocal
These cancel each other out.
They can be freely moved through the integral sign because they do not involve the integration variable ##k##.
EDIT: Darn, Dr Claude jinxed me. Now I'm not allowed to talk for the rest of the day.
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