Question

نشان دهید: $$ \int _0^ \infty lnx Arctan ( \frac{1}{ x^{2} } )dx= \frac{- \pi }{4 \sqrt{2} } (4- \pi )$$

Answer 1

$$ i\int _0^ \infty lnx Arctan ( \frac{1}{ x^{2} } )=i \frac{ \pi }{4 \sqrt{2} } ( \pi -4) \Longrightarrow $$ return $$ \int _0^ \infty lnxln(1+ \frac{1}{ x^{4} } )dx= -\frac{ \pi }{2 \sqrt{2} }(4+ \pi ) $$