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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.