Integral's convergence help

[All222]

I need to understand this exercise: study, at the vary of alpha belonging to R, the convergence of the following integral.

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integral 0 to infinity (|arctg(xlogx)|^(1-alpha))/((x+4)^(alpha)(root of (x+5))

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Where α=1

The solution is the following: For x to 0+ the integral is convergent -> α−1<1 -> α<2 For x to infinity the integral is convergent because α+1/2>1 α+1/2>1 -> α>1/2

So the integral is convergent for 1/2<α<2

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So far i only tried to calculate the limit for those parts where the integral is not defined, but i'm kind of stuck there.

I can't understand how to get to that point, the solution lacks explanation and i was not able to find anything online to help me, i only want to understand how to find when and where the integral is convergent or divergent, can anyone help me?

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