I am working on the following problem:
Determine the values of $p$ and $q$ for which the integral $\int_0^{\infty} \frac{x^{p-1}}{1+x^q}dx$ exists as a Lebesgue integral.
This is what I thought. There are potential issues at $0$ and $\infty$, so:
- at $0$, $\frac{x^{p-1}}{1+x^q} \sim x^{p-1}$, so I need $p-1>-1$, which gives $p>0$;
- at $\infty$, $\frac{x^{p-1}}{1+x^q} \sim x^{p-1-q}$, so I need $p-1-q<-1$, which gives $p-q<0$.
In conclusion, the two inequalities above give a range of values for $p$ and $q$. Is my work correct? I appreciate every comment/correction.