In Folland's book on real analysis he claims that the function $f_\alpha(x) = x^{-\alpha}$ is in $L^p((0,1))$ if $\alpha > 0$ and $\alpha p < 1$.
I tried to verify this directly: $$\left[ \int_{(0,1)} |x^{-\alpha}|^pdx\right]^{1/p} = \left[ \int_{(0,1)} x^{-\alpha p}dx\right]^{1/p} \leq \left[ \int_{(0,1)} x^{-1}dx\right]^{1/p} = \left[ \log(1) - \log(0)\right]^{1/p} = \left[- \log(0)\right]^{1/p}$$ or another approch: $$\left[ \int_{(0,1)} |x^{-\alpha}|^pdx\right]^{1/p} = \left[ \int_{(0,1)} x^{-\alpha p}dx\right]^{1/p} = \left[\left(\frac{1}{(x^{1-\alpha p})(1-\alpha p)}\right)\Big\vert_{x = 0}^{x=1}\right]^{1/p}.$$
However, in both cases the final expression will not be bounded, so how is this function in $L^p$? Based on the wording in the book I assume this should be trivial, so I must be overlooking something.