How to solve this integration: int_0^1 x^(2012)/(1+e^x)dx

Kyler Oconnor

Kyler Oconnor

Answered question

2022-11-14

How to solve this integration: 0 1 x 2012 1 + e x d x

Answer & Explanation

luluna81mxmbk

luluna81mxmbk

Beginner2022-11-15Added 17 answers

One approach would be to write
1 1 + e x = 1 e x ( 1 + e x ) = e x k = 0 ( 1 ) k e k x = k = 0 ( 1 ) k e ( k + 1 ) x 0
with uniform convergence on x c for every c > 0. Hence
0 1 x 2012 1 + e x d x = 0 1 ( k = 0 ( 1 ) k e ( k + 1 ) x x 2012 ) d x = k = 0 ( ( 1 ) k 0 1 e ( k + 1 ) x x 2012 d x ) = k = 0 ( ( 1 ) k 0 k + 1 e t ( x k + 1 ) 2012 1 k + 1 d t ) = k = 0 ( 1 ) k ( k + 1 ) 2013 γ ( 2013 , k + 1 )

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