Takeover Season
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Can someone please help on this convergence, it is pretty hard! Thanks a lot!
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RDKGames
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Can someone please help on this convergence, it is pretty hard! Thanks a lot!
Note that

\displaystyle \int_0^{\infty} \dfrac{x^\alpha \arctan x}{2 + x^4} \ dx = \int_0^{1} \dfrac{x^\alpha \arctan x}{2 + x^4} \ dx + \int_1^{\infty} \dfrac{x^\alpha \arctan x}{2 + x^4} \ dx

When is the first integral over (0,1) is convergent?

For the second integral over (1,\infty), notice that for large x the integrand behaves like \dfrac{1}{x^{4-\alpha}} because \arctan (x) \sim 1.
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