dilogarithm-integral-calculate-0-1-li-2-1-x-2-dx- Tinku Tara June 3, 2023 Differentiation 0 Comments FacebookTweetPin Question Number 135055 by mnjuly1970 last updated on 09/Mar/21 ….dilogarithmintegral….calculate:::\boldsymbolϕ=∫01li2(1−x2)dx=? Answered by Ñï= last updated on 10/Mar/21 ∫01Li2(1−x2)dx=xLi2(1−x2)∣01−∫014x2⋅lnx1−x2dx=4∫01(1−11−x2)lnxdx=4∫01lnxdx−4∫01lnx1−x2dx=4(xlnx−x)∣01−4[12ln1+x1−xlnx∣01−12∫01ln(1+x)−ln(1−x)xdx]=−4+2∫01ln(1+x)−ln(1−x)xdx=−4+2[−Li2(−1)+Li2(1)]=−4+3Li2(1)=π22−4 Commented by mnjuly1970 last updated on 10/Mar/21 veryniceverynicewiththethanking… Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Define-the-sequence-a-n-by-the-recursive-formula-a-n-1-ca-n-nr-n-1-n-Z-n-1-with-a-1-h-and-c-r-h-C-c-r-0-Find-a-n-in-terms-of-n-Next Next post: S-0-sin-e-x-cos-e-x-dx-1-Solve-S-Also-Does-2-0-sin-e-x-cos-e-x-dx- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.
![∫_0 ^1 Li_2 (1−x^2 )dx=xLi_2 (1−x^2 )∣_0 ^1 −∫_0 ^1 4x^2 ∙((lnx)/(1−x^2 ))dx =4∫_0 ^1 (1−(1/(1−x^2 )))lnxdx=4∫_0 ^1 lnxdx−4∫_0 ^1 ((lnx)/(1−x^2 ))dx =4(xlnx−x)∣_0 ^1 −4[(1/2)ln((1+x)/(1−x))lnx∣_0 ^1 −(1/2)∫_0 ^1 ((ln(1+x)−ln(1−x))/x)dx] =−4+2∫_0 ^1 ((ln(1+x)−ln(1−x))/x)dx =−4+2[−Li_2 (−1)+Li_2 (1)] =−4+3Li_2 (1) =(π^2 /2)−4](https://www.tinkutara.com/question/Q135107.png)
