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PF PotW Robot
Here is this week's advanced math problem of the week. We have several members who will check solutions, but we also welcome the community in general to step in. We also encourage finding different methods to the solution. If one has been found, see if there is another way. Occasionally there will be prizes for extraordinary or clever methods. Spoiler tags are optional.
Using contour integration, prove
$$\int_0^\infty \sin(x^\alpha)\, dx = \sin\left(\frac{\pi}{2\alpha}\right)\,\Gamma\!\left(1 + \frac{1}{\alpha}\right),\quad \alpha > 1.$$
(PotW thanks to our friends at http://www.mathhelpboards.com/)
Using contour integration, prove
$$\int_0^\infty \sin(x^\alpha)\, dx = \sin\left(\frac{\pi}{2\alpha}\right)\,\Gamma\!\left(1 + \frac{1}{\alpha}\right),\quad \alpha > 1.$$
(PotW thanks to our friends at http://www.mathhelpboards.com/)