$1.($a$)$ Show that f$($x$)=$ H$($x$)$ ln x is locally integrable on R$.($The definition of f means

that f$($x$)=0$ for x $0$ and f$($x$)=$ ln x for x $>0.)$

$($b$)$ Is f$($x$)=$ H$($x$)1$

x locally integrable on R$?$ Justify your answer.