Some complex Fourier-like infinite integral

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Homework Statement


The following integral is given:
[tex]\int_{0}^{\infty} e^{ikx} dx[/tex]
k & x are real.

Homework Equations


We know that:
[tex]\int_{-\infty}^{\infty} e^{ikx} dx=2\pi \delta(k)[/tex]

The Attempt at a Solution


The answer is:
[tex]\mathcal{I}=\pi \delta(k) + i \frac{1}{k}[/tex]
I could easily prove the real part with the formula in 2, but couldn't be persuaded why the imaginary part is so (it barely makes sense). I tried:
[tex]\Im \int_{0}^{\infty} e^{ikx} dx = \int_{0}^{\infty} =\sin(kx) dx = \lim_{M\rightarrow \infty} \frac{1}{k}(-\cos(kM)+1)[/tex]
But [itex]\lim_{M\rightarrow \infty} \cos(kM)[/itex] is quite meaningless and I can't see why the whole expression should be k^{-1}...

I'll appreciate any help,
Thanks!
 
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