Im stuck solving the following integral: $$\int \limits_{-1}^{3} \frac{(x^2 - 3x - 4)^{\sqrt{a}}}{\arctan^{2a}\left(\sqrt{2 + x} - 1\right)}dx$$
I think at the beginning I should use the property about equivalent functions (if we change some function for the equivalent one, both the initial and resulted integrals will converge or diverge together, sorry for sorry for such an informal explanation).
So I change $\arctan^{2a}(\sqrt{2 + x} - 1)$ to $(\sqrt{2 + x} - 1)^{2a}$, and then I don't know what to do next...
Could you please tell me if I'm doing something wrong, or could you please give some hints on how to solve this (show for which $a$ this integral converges and for which $a$ it diverges).
hint
Put $x+1=t$
use $$\sqrt {1+t}-1\sim t/2 \;\;(t\to 0)$$ $$\arctan (X)\sim X \;\;(X\to 0) $$ the integrand is equivalent to
$$K\frac {t^\sqrt {a}}{t^{2a}}$$
and the integral converges if and only if
$$2a-\sqrt {a}<1$$ $$\iff 0\le a <1$$