nice-calculus-0-1-ln-1-x-2-x-2-1-dx-pi-2-24-NOTE-li-2-z-li-2-1-z-pi-2-6-ln-z-ln-1-z-Hence-

Question Number 139478 by mnjuly1970 last updated on 27/Apr/21

\dots \dots \dots n i c e \dots \dots \dots c a l c u l u s \dots \dots . .

Φ := \int_{0}^{1} \frac{l n (\frac{1 + x}{2})}{x^{2} - 1} d x = \frac{π^{2}}{24}

N O T E :: {l i}_{2} (z) + {l i}_{2} (1 - z) = \frac{π^{2}}{6} - l n (z) l n (1 - z)

H e n c e :: {l i}_{2} (\frac{1}{2}) = \frac{π^{2}}{12} - \frac{1}{2} {l n}^{2} (2)

Φ : \overset{⟨ \frac{1 + x}{2} = y ⟩}{=} 2 \int_{\frac{1}{2}}^{1} \frac{l n (y)}{4 y^{2} - 4 y} d y

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

:= - \frac{1}{2} {[\frac{1}{2} {l n}^{2} (y)]}_{\frac{1}{2}}^{1} + \frac{1}{2} {{l i}_{2} (1) - {l i}_{2} (\frac{1}{2})}

:= \frac{1}{4} {l n}^{2} (2) + \frac{π^{2}}{12} + \frac{1}{2} (- \frac{π^{2}}{12} - \frac{1}{2} {l n}^{2} (2))

Φ := \frac{π^{2}}{24}