I am studying an integral using comparison text. I have managed to show it easily that for $b < 1$, $$\int_0^1 \frac{ln(1+x)}{x^b}dx$$ convergens and this is because I am well aware of functions that behave a certain way when $b$ is either less than one or greater than one.
But indeed, this integral is also convergent for $b < 2$. What could I possibly compare with to show that? Or is there a method that does not require the comparison text?
I just began studying this, so my repertoire of functions to compare with is very limited. I basically know little else but $\int_a^b \frac{1}{x^b} dx$ (which is what I used for the case $b \le 1$)