Is $\int_0^1 \frac{x^p}{1＋x^q} \;dx < \infty$?

Question

When $p>-1, \;q>0$, is the following true?

$$\int_0^1 \frac{x^p}{1＋x^q}\; dx < \infty$$

Any help would be appreciated. I observed by graph soft this may be true, but I’m not certain. Thank you.

Answer 1

Hint

If $p>-1$ and $q>0$ then, if $x\in (0,1)$,

$$\frac{x^p}{1+x^q}=\frac{x^{p-q}}{x^{-q}+1}<x^{p-q}.$$

Answer 2

If $q>0$ and $x \in [0,1]$, then $1 \le 1+x^q \le 2$, so $$ \int_0^1\frac{x^p}{1+x^q}\;dx $$ converges if and only if $$ \int_0^1 x^p\;dx $$ converges. That is, if $p>-1$.

Answer 3

With Asymptotic Analysis:

Near $0$, $\:1+x^q\sim 1$, hence $$\frac{x^p}{1+x^q}\sim_0 x^p,$$ and $\displaystyle\int_0^1\!x^p\,\mathrm dx<\infty$ if $p>-1$.