How to integrate int log(e^x+e^(-x))dx ?

Brenda Jordan

Brenda Jordan

Answered question

2022-11-03

How to integrate log ( e x + e x ) d x

Answer & Explanation

Zoey Benitez

Zoey Benitez

Beginner2022-11-04Added 18 answers

For x>0, we have
log ( e x + e x ) = log ( e x ( 1 + e 2 x ) ) = x + log ( 1 + e 2 x ) = x + k = 1 ( 1 ) k 1 e 2 k x k
We now have
I = 0 a log ( e x + e x ) d x = 0 a ( x + k = 1 ( 1 ) k 1 e 2 k x k ) d x = a 2 2 + k = 1 ( 1 ) k 1 k 0 a e 2 k x d x = a 2 2 + k = 1 ( 1 ) k 1 k 1 e 2 k a 2 k = a 2 2 + π 2 24 + 1 2 k = 1 ( e 2 a ) k k 2 = a 2 2 + π 2 24 + Li 2 ( e 2 a ) 2

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