I'm struggling to analyze an improper integral of two variables.
$$ \lim_{y\rightarrow \infty}\int_1^y f(x)\sqrt{y-x}dx $$ where $f$ is a real valued function defined on $[0,\infty)$.
One simple example $f(x)= 1$ does not make the integral converge.
I want to find some sufficient condition for the improper integral to be well defined.
My expected answer is like
$$ \int_1^{\infty} \frac{1}{x^p} dx $$ is convergent $p>1$ and divergent $p<=1$.
Thank you for advance.
My question may be somewhat vague. Ask me freely for some points making you confused. Thank you!