Spoiler:

Let denote our integral. Then, evidently

Switch the order of integration (Fubini's theorem is satisfied) to get

Rewrite this as

Make the obvious substitution ( ) on the inside to turn it into

Thus, the first integral being easy, and the second being the definition of the Euler-Mascheroni constant we find that

To prove the second assertion merely write the exponentials in complex notation and separate the real and imaginary parts. The theorem then becomes immediate from the common theorem that