Could you help me understand the correct way to see the behavior of these 2 integrals?
$$\int_{0}^{π/2} \frac {dx}{\tan(x)} $$
In 0: 1/tan(x) ~ 1/x (diverge)
In π/2: is it right to use a direct substitution? Like 1/tan(π/2)=number (converge)
$$\int_{0}^{1} \frac {dx}{ \log{(x-x^2)} }$$
In 0: ~ 1/logx
In1: I don't know
Note that
$$\int_{0}^{π/2} \frac {dx}{\tan(x)}=\int_{0}^{π/4} \frac {dx}{\tan(x)}+\int_{π/4}^{π/2} \cot x\,dx$$
and the first integral diverges for limit comparison with $\frac1x$.
Note that the second part converges but it is not influent on the result. For the divergence it suffices to note that the first diverges.
For the second note that for $x\to 0^+$ and for $x\to 1^-$
$$\frac {1}{ \log{(x-x^2)}}\to 0$$
thus the integral converges (since the integrand is continuos and bounded on the interval).