# Convergence of an improper integral of two variables

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.

$$\int_1^{\infty} \frac{1}{x^p} dx$$ is convergent $p>1$ and divergent $p<=1$.
If $f$ is any positive measurable function then the limit is $\infty$ by Monotone Convergence Theorem.
• Thank you for your reply. But, $f$ is not restricted to a positive function. :< – user155214 Jun 5 '18 at 6:47