Some complex Fourier-like infinite integral

[Ahmes]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

The answer is:
$$\mathcal{I}=\pi \delta(k) + i \frac{1}{k}$$
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:
$$\Im \int_{0}^{\infty} e^{ikx} dx = \int_{0}^{\infty} =\sin(kx) dx = \lim_{M\rightarrow \infty} \frac{1}{k}(-\cos(kM)+1)$$
But $\lim_{M\rightarrow \infty} \cos(kM)$ is quite meaningless and I can't see why the whole expression should be k^{-1}...

I'll appreciate any help,
Thanks!

 

[Ahmes]

Never mind, I wasn't the first to be upset by it:
http://www.dms.uaf.edu/~bueler/M611heaviside.pdf

But giving such thing in QED in plain wrong.