What is a solution to this integral? What is a solution to the following integral:
$$\int_0^\infty \, \frac{\cos{kt}}{\pi}\,\mathrm{d}k\,?$$
I have tried to evaluate this in the usual way:
$$\begin{align}\frac{1}{\pi}\int_0^\infty\,\cos{kt}\,\mathrm{d}k &= \frac{1}{\pi t} \left[\sin{k t}\right]\bigg|_{k\rightarrow 0}^{k\rightarrow \infty} \\ &=\frac{1}{\pi t} \left[\lim_{k\rightarrow \infty}\sin{k t} - \sin 0\right]  \\ &=\frac{1}{\pi t} \lim_{k\rightarrow \infty}\sin{k t},\end{align}$$
but $\lim_{k\rightarrow \infty}\sin{k t}$ doesn't converge within infinity. 
What is the trick here? Note that this is a homework question, so I don't need the complete solution, just directions where to start.

The integral doesn't converge to any finite value. A graph of the function you're integrating would be a transformed version of $\cos k$, which oscillates back and forth. The area underneath would also oscillate back and forth, which is what your answer already tells you.
