prove a function is not Lebesgue integrable

Let $ f$ be a function defined on the interval $[a, b]$, and thereexist constants $ M > 0 $ and $ \alpha \geq 1 $ such that

$$ |f(x)| \geq \dfrac{M}{|x - x_0|^\alpha} $$

for some $ a < x_0 < b $, for almost all $ x \in \mathbb{R} $. Then $ f $ is not Lebesgue integrable.

My idea is to prove that $\dfrac{M}{|x - x_0|^\alpha}$ is not Lebesgue integrable. Since then, $|f|$ is not Lesbegue integrable and we already know that $|f|$ is L-integrable $\iff f$ is L-integrable, hence $f$ is not L-integrable.
But currently, I don't know how to prove $\dfrac{M}{|x - x_0|^\alpha}$ is not Lebesgue integrable (Also, I was hinted to use Dominated convergence theorem). Any help I would be very grateful!