Question Number 114975 by mnjuly1970 last updated on 22/Sep/20

              ....  nice  mathematics....            show  that ::                                    Φ = ∫_0 ^( 1) li_2 (x)dx = (π^2 /6) −1  ✓                          m.n.july.1970

Answered by maths mind last updated on 24/Sep/20

li_2 (x)=Σ(x^n /n^2 )  ∫_0 ^1 li_2 (x)=Σ_(n≥1) ∫_0 ^1 (x^n /n^2 )dx=Σ_(n≥1) (1/((n+1)n^2 ))  =Σ((1/n^2 )+(1/(n+1))−(1/n))  =Σ_(n≥1) (1/n^2 )+Σ_(n≥1) ((1/(n+1))−(1/n))=(π^2 /6)−1