Here's more divergence craziness:
$$\int_0^\infty \sin\;u\mathrm{d}u=1$$
and
$$\int_0^\infty \ln\;u\;\sin\;u\mathrm{d}u=-\gamma$$
($\gamma$ is of course the Euler-Mascheroni constant)
which only makes sense when interpreted as
$$\lim_{\varepsilon\to 0} \int_0^\infty \exp(-\varepsilon u)\ln\;u\;\sin\;u\mathrm{d}u$$
and similarly for the first one.
Results obtained from numerical quadrature methods specially designed for infinite oscillatory integrals (e.g. the Ooura-Mori double exponential quadrature and the Longman scheme) agree with these closed forms.
