Improper integral with sine and cosine

Question

How does this integral behave with respect to $\alpha$ parameter? $$\int_1^{3/2} \frac{(1-\cos^2(\pi x))^{\alpha}}{\sin(\sqrt{x^2-1})} \ dx$$ I am really unable to say anything constructive about this one..

Answer 1

First observation is that by changing the variable $t=x-1$ we get $$\int_1^{3/2} \frac{(1-\cos^2(\pi x))^{\alpha}}{\sin(\sqrt{x^2-1})} \ dx=\int_0^{1/2}\frac{\sin^{2\alpha}(\pi t)}{\sin\left(\sqrt{t(t+2)}\right)}\, dt=\int_0^{1/2}f_\alpha(t)\, dt$$ It is easy to check that $\lim_{t\to0}f_\alpha(t)$ exists iff $\alpha\geq\frac14$.
So, for $\alpha\geq\frac14$ we have a proper integral of a continuous function.

Answer 2

It diverges for $a<0$, and converges otherwise $($in absolute value$)$, being strictly decreasing $($again, in absolute value$)$ for $a\ge0$, due to the fact that the sine function is always $\le1$ in absolute value. Sign-wise, we have negative values for all $a\in\mathbb N$, as well as for rational a with odd denominators in simplified form. Otherwise, the values are complex, i.e., when a is either irrational, or rational with even denominators in simplified form.