Nice-Mathematics-calculation-0-1-tanh-1-x-x-dx-pi-2-4-x-t-2-0-1-tanh-1-t-t-

Question Number 157251 by mnjuly1970 last updated on 21/Oct/21

You can't use 'macro parameter character #' in math mode

\dots c a l c u l a t i o n \dots

Ω := \int_{0}^{1} \frac{{t a n h}^{- 1} (\sqrt{x})}{x} d x \overset{?}{=} \frac{π^{2}}{4}

- - - - - - - - - - - - -

Ω : \overset{\sqrt{x} = t}{=} 2 \int_{0}^{1} \frac{{t a n h}^{- 1} (t)}{t} d t

: \overset{{{t a n h}^{- 1} (t) = \frac{1}{2} l n (\frac{1 + t}{1 - t})}}{=} \int_{0}^{1} \frac{l n (1 + t) - l n (1 - t)}{t} d t

: = - {Li}_{2} (- 1) + {Li}_{2} (1)

: \overset{{{Li}_{2} (z) = \underset{n = 1}{\sum^{\infty}} \frac{z^{n}}{n^{2}}}}{=} η (2) + ζ (2)

:= \frac{π^{2}}{12} + \frac{π^{2}}{6} = \frac{π^{2}}{4} m . n

Answered by qaz last updated on 21/Oct/21

\int_{0}^{1} \frac{\tanh^{- 1} (\sqrt{x})}{x} dx

= 2 \int_{0}^{1} \frac{\tanh^{- 1} x}{x} dx

= 2 \underset{n = 0}{\sum^{\infty}} \frac{1}{2 n + 1} \int_{0}^{1} x^{2 n} dx

= 2 \underset{n = 0}{\sum^{\infty}} \frac{1}{{(2 n + 1)}^{2}}

= 2 (1 - 2^{- 2}) ζ (2)

= \frac{π^{2}}{4}

Commented by mnjuly1970 last updated on 21/Oct/21

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