pls-how-to-calculate-this-1-2-1-ln-x-1-x-dx-

Question Number 205264 by pticantor last updated on 13/Mar/24

\boldsymbol p l s \boldsymbol h o w \boldsymbol t o \boldsymbol c a l c u l a t e \boldsymbol t h i s ?

\int_{\frac{1}{2}}^{1} \frac{\boldsymbol l n (\boldsymbol x + 1)}{\boldsymbol x} \boldsymbol d x

Answered by Berbere last updated on 13/Mar/24

= - (- \int_{\frac{1}{2}}^{1} \frac{l n (1 - (- x))}{(- x)} d (- x))

= - {[{L i}_{2} (- x)]}_{\frac{1}{2}}^{1} = - {L i}_{2} (- 1) + {L i}_{2} (- \frac{1}{2})

= \frac{π^{2}}{12} + {L i}_{2} (- \frac{1}{2})

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