How do I show the following: ∫ 0 α d x ∫ 0 α d...

Jaqueline Kirby

Jaqueline Kirby

Answered

2022-06-26

How do I show the following:
0 α d x 0 α d y ln | x y | x y = 4 α ( ln ( α ) + 2 ln ( 2 ) 3 )
where α > 0.

Answer & Explanation

Harold Cantrell

Harold Cantrell

Expert

2022-06-27Added 21 answers

Proceed as follows
0 a 0 a ln | x y | x y d y d x = 2 0 a 0 x ln ( x y ) x y d y d x = y = x t 2 4 0 a 0 1 ( ln x + ln ( 1 t 2 ) ) d t d x = 4 0 a ( ln x + 2 ln 2 2 ) d x = 4 a ( ln a + 2 ln 2 3 )

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