Show ∫ s ∞ f ( x ) d x = L { F (...

undergoe8m

undergoe8m

Answered

2022-11-21

Show s f ( x ) d x = L { F ( t ) t } given f ( x ) = 0 e x t F ( t ) d t

Answer & Explanation

apopihvj

apopihvj

Expert

2022-11-22Added 20 answers

Observe that
e s t t = s e x t d x .
Then
0 e s t F ( t ) t d t = 0 s e x t d x F ( t ) d t = s 0 e x t F ( t ) d t d x = s f ( x ) d x ,
where in the second equality I have assumed that F is nice enough so that you can exchange the order of integration.

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