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polivijuye

polivijuye

Answered question

2022-06-11

Integrating ln ( x ) × ln ( 1 x )
Is there a way I can derive the value of the integral
0 1 ln ( x ) ln ( 1 x ) d x
using the fact that
i = 1 1 n 2 = π 2 6
? (the actual value of the integral is 2 π 2 6 )
Thanks in advance

Answer & Explanation

Amy Daniels

Amy Daniels

Beginner2022-06-12Added 20 answers

Hint: In addition to previous comments:
ln ( 1 x ) = i = 1 x i i
and
x i ln ( x ) d x = 1 ( i + 1 ) 2
and
i = 1 1 i ( i + 1 ) 2 = i = 1 1 i ( i + 1 ) 1 ( i + 1 ) 2
Llubanipo

Llubanipo

Beginner2022-06-13Added 9 answers

You could be interested by the fact that the result of 0 a ln ( x ) ln ( a x ) d x is given by
1 6 a ( 6 ( log ( a ) 2 ) log ( a ) + π 2 12 )
which is not much more difficult to establish than in the case where a = 1

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