I want to determine the values of $a$ and $b$ for which the following integrals converge:
(i) $$\int \limits _0 ^{\infty} \frac{x^{a-1}}{1+xb}dx$$
(ii) $$\int \limits _0^1 x^a(1-x^2)^b dx$$
My thought process thus far is as follows:
For (i), I thought about using the Comparison Test and the fact that $\int _1^{\infty}\frac{1}{x^p}dx$ diverges if $p \le 1$ and converges if $p > 1$. Thus, I split the integral in (i) into 2:
$$\int \limits _0 ^{\infty} \frac{x^{a-1}}{1+xb}dx = \int \limits _0 ^1 \frac{x^{a-1}}{1+xb}dx + \int \limits _1 ^{\infty} \frac{x^{a-1}}{1+xb}dx$$ If I can find that both parts converge, the improper integral should converge. For the second part of the integral, note that
$$\frac{x^{a-1}}{1+xb} \le \frac{x^{a-1}}{x} = x^{a-2}, \quad \forall x\in (1,\infty)$$ when $b \ge 1$. Moreover, $$\int \limits _1 ^{\infty} x^{a-2} dx$$ converges when $a < 1$. Then, by the Comparison Test, does the second part of the integral converge when $a < 1$ and $b \ge 1$? Are these limits for $a$ and $b$ exhaustive? Also, does the first part of the integral converge? How would I check that? Do I use integration by parts?
For (ii), if both $a \ge 0 $ and $b \ge 0$, the integral converges, as the function is either a constant or a polynomial. From examination, I believe that the degree of the product has to be greater than -1; that is, $a+ 2b > -1$, as I think $\int \limits _0 ^1 x^p dx $ converges if $ p > -1$ and diverges if $p \le -1$, but I'm not sure how to prove that rigorously.
Any guidance would be appreciated. Thank you in advance!