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Question Number 156617 by mnjuly1970 last updated on 13/Oct/21

Answered by mindispower last updated on 14/Oct/21

e^(−x) =t =∫_0 ^1 ((ln(1+t)(1−t))/(1+t)).(dt/t) =∫_0 ^1 (−((2ln(1+t))/(1+t))+((ln(1+t))/t))dt =−ln^2 (2)−Li_2 (−1)=(π^2 /(12))−ln^2 (2)=(((π^2 /(12))−ln^2 (2))/((π^2 /(12))−((ln^2 (2))/2))).Li_2 ((1/2)) Li_2 ((1/2))=(π^2 /(12))−((ln^2 (2))/2)

e−x=t=∫01ln(1+t)(1−t)1+t.dtt=∫01(−2ln(1+t)1+t+ln(1+t)t)dt=−ln2(2)−Li2(−1)=π212−ln2(2)=π212−ln2(2)π212−ln2(2)2.Li2(12)Li2(12)=π212−ln2(2)2

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