Construction of an integrable function with a function in $L^2$ I have this really simple question, but I cannot figure out the answer. Suppose that $f\in L^2([0,1])$. Is it true that $f/x^5$ will be in $L^1([0,1])$? 
Thanks! 
Edit: I was interested in $f/x^{1/5}$.

The statement

$$f \in L^2([0,1])\Rightarrow fx^{-\frac{1}{5}} \in L^1([0,1])$$

is true. It is an easy application of Hölder inequality. Infact, by hypothesis $f\in L^2([0,1])$; on the other hand, we have $g(x)=x^{-1/5}\in L^2([0,1])$ so by Hölder (indeed, this is the case "Cauchy-Schwarz") 
$$
\Vert fg \Vert_1 \le \Vert f\Vert_2 \Vert g \Vert_2
$$
hence $fg\in L^1$ (and you get also an upper bound for its $L^1$-norm).
