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0-1-2-ln-1-t-ln-t-t-dt-I-m-about-to-give-up-

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Question Number 109136 by EmericGent last updated on 21/Aug/20
∫_0 ^(1/2) ((ln(1-t)ln(t))/t) dt I′m about to give up
∫01/2ln(1−t)ln(t)tdtI′mabouttogiveup
Answered by Sarah85 last updated on 21/Aug/20
∫((ln (1−t) ln t)/t)dt by parts f′=((ln (1−t))/t) ⇒ f=−Li_2 t g=ln t ⇒ g′=(1/t) −Li_2 t ln t −∫−((Li_2 t)/t)dt= =Li_3 t −Li_2 t ln t +C ⇒ answer is (7/8)ζ(3)−(1/3)(ln 2)^3
∫ln(1−t)lnttdtbypartsf′=ln(1−t)t⇒f=−Li2tg=lnt⇒g′=1t−Li2tlnt−∫−Li2ttdt==Li3t−Li2tlnt+C⇒answeris78ζ(3)−13(ln2)3

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