Show that $f$ is Lebesgue integrable I am given a function defined on $(\mathbb R,$ Borelians, Lebesgue measure)
such that 
$$f= \begin{cases}  + \infty &amp;\text{if }  x=0 \\ 
\ln(|x|) &amp;\text{if } 0\lt |x|\lt1 \\
0 &amp;\text{if } |x|\ge1 \end{cases}
$$


  How can I show that this function is Lebesgue-integrable ?
Since $f$ is even it is enough to show that it is integrable on $(0,1)$. But $$\int_{(0,1)} f(u)du =ulnu-u|_0^1 =1$$
Since $f$ is even it is enough to show that it is integrable on $(0,1)$. But $$\int_{(0,1)} f(u)du =ulnu-u|_0^1 =1$$