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Lillianna Andersen 2022-07-07 Answered
Prove that 0 x n e x n d x 0 e x n d x = 1 n
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Answers (2)

Johnathan Morse
Answered 2022-07-08 Author has 18 answers
It suffices to prove that
0 ( x n 1 / n ) e x n d x = 0
or
0 [ 1 n x e x n ] d x = 0
or
0 [ 1 n x e x n ] d x = 0
or
[ 1 n x e x n ] | 0 = 0
which is true.
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Savanah Boone
Answered 2022-07-09 Author has 5 answers
Consider
0 x α e x β d x
with α 0 , β > 0 . Letting y = x β gives
0 x α e x β d x = 1 β 0 y α + 1 β 1 e y d y = 1 β Γ ( α + 1 β )
Where Γ is the Euler-Gamma function. Hence,
0 x n e x n d x 0 e x n d x = Γ ( 1 n + 1 ) Γ ( 1 n ) = 1 n
for all n > 0
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