Can the $p$-test be used to determine convergence of integrals of the form $\int^b_a \frac{1}{(c-f(x))^p}dx$?

Question

Can the $p$-test be used to determine convergence of integrals of the form $\int^b_a \frac{1}{(c-f(x))^p}dx$ where $c>f(x)$ for $a\leq x < b$ and $f(b)=c$?

Answer 1

Hint: For $x$ near $b,$

$$ 2|f'(b)| > \left | \frac{f(x)-f(b)}{x-b}\right | = \left| \frac{f(x)-c}{x-b}\right| > \frac{|f'(b)|}{2}.$$