I didnt want to post all of them in one question so i split my question into 2 parts. Basically i just can't get my head around how to prove whether or not these converge or diverge i have been thus far just guess which it does and then trying to prove it but it went very poorly for me on my last exam. if someone could show me what function or method they used to prove that the function converges or diverges for the following i would very much appreciate it.
1)$\int^{1}_{0} \frac{xdx}{1-x^{2}}$
2)$\int^{\pi}_{\pi/2} \cot (x) dx$
3) $\int^{1}_{0} \frac{(1-x)dx}{x^{2}-4x+3}$
4)$\int^{1}_{0} \frac{dx}{x^{1/2}(x^{2}+x)^{1/3}}$
Hints:
1) The change of variable $x=1-t$ would bring the badness to $0$, which is more familiar territory.
2) I suggest letting $x=\pi -t$. This will bring the badness to $0$. Now use the fact that for $t$ near $0$, we have $\frac{\sin t}{t}\approx 1$.
3) There is much less to this than meets the eye: factor.
4) Rewrite the bottom as $x^{1/2}(x^{1/3})(x+1)^{1/3}=x^{5/6}(x+1)^{1/3}$. This means that your integrand behaves like $\frac{1}{x^{5/6}}$ near $0$. It is in fact $\le \frac{1}{x^{5/6}}$ on our interval. But $\int_0^1 \frac{dx}{x^{5/6}}$ converges, since $5/6\lt 1$. So by Comparison our integral converges.