I need to understand this exercise from my math book: study, at the vary of alpha belonging to R, the convergence of the following integral and then calculate the value for $a = 1$.
$$ \int_0^{4} \frac{ \left( 5+3\sqrt{x} \right)(\arctan x)^{2a-1} }{(4x-x^{2})^{\alpha} } \, dx $$
Where $\alpha=\frac{1}{2}$
The solution is the following: For $x$ to $0^+$ the integral is convergent $\Rightarrow$ For $x$ to $\infty$ the integral is convergent because $\frac{5}{4^{a}x^{1-a}(1+o(1))}$ $\Rightarrow$ the integral is convergent for $a$ $>$ $0$
For $x$ to ${4^{-}}$ the integral is convergent $\rightarrow$ $\frac{10(\arctan4)^{2a-1}(1+o(1))}{2^{a}(4-x)^{a}(1+o(1))}$
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? I got to the point where I know that I need to calculate the limit where the integral is undefined, but I'm kind of stuck there. Can someone help me understand the solution?