My teacher proposed us the following challenge:
Provide conditions for the reals numbers $a$ and $b$ such that the following integral converges and compute the value:
$$ \int_0^c \left|\frac{1}{t^a}\right|^b dt, $$ where $c>0$ but fix.
He mentioned that the integrand is well defined for some Lebesgue's spaces.
Already I tried to compute formally, obtaining:
$$ \int_0^c \left|\frac{1}{t^a}\right|^b dt= \int_0^c \left(\frac{1}{t^a}\right)^b dt= \left. \frac{t^{-ab+1}}{-ab+1} \right|_0^c = \frac{c^{-ab+1}}{-ab+1}. $$
Do you have some advice?
Thanks