How is natural log integration broken up into this range? (equation is contained the script) When I

gledanju0

gledanju0

Answered question

2022-06-26

How is natural log integration broken up into this range? (equation is contained the script)
When I was reading a paper, I found an strange derivation like
+ l n ( 1 + e w ) f ( w ) d w = 0 ln ( 1 + e w ) f ( w ) + 0 [ ln ( 1 + e w ) + w ] f ( w ) d w
when w is the normal random variable and f ( w ) is the normal density.
Why is that natural log integration broken up into that?
I think it just need a simple principle.
Thank you.

Answer & Explanation

Trey Ross

Trey Ross

Beginner2022-06-27Added 30 answers

Note that 1 + e w = 1 + 1 e w = 1 + e w e w
Taking the ln, we obtain ln ( 1 + e w ) = ln ( 1 + e w e w ) = ln ( 1 + e w ) ln ( e w ) = ln ( 1 + e w ) w
Therefore ln ( 1 + e w ) = ln ( 1 + e w ) + w
It follows that 0 ln ( 1 + e w ) f ( w ) d w = 0 ( ln ( 1 + e w ) + w ) f ( w ) d w
Emmy Knox

Emmy Knox

Beginner2022-06-28Added 10 answers

ln ( 1 + e w ) = ln ( e w ( e w + 1 ) ) = ln ( e w ) + ln ( 1 + e w ) = ln ( 1 + e w ) + w

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