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Question Number 160636 by mnjuly1970 last updated on 03/Dec/21
prove that ∫_(0 ) ^( 1) (((tanh^( −1) ( x ))/x) )^( 2) = ζ ( 2 ) ■ m.n
provethat∫01(tanh−1(x)x)2=ζ(2)◼m.n
Answered by Kamel last updated on 03/Dec/21
=^(t=((1−x)/(1+x))) (1/2)∫_0 ^1 ((Ln^2 (t))/((1−t)^2 ))dt=^(IBP) −∫_0 ^1 ((Ln(t))/((1−t)))dt=ζ(2)
=t=1−x1+x12∫01Ln2(t)(1−t)2dt=IBP−∫01Ln(t)(1−t)dt=ζ(2)
Commented by mnjuly1970 last updated on 04/Dec/21
mercey sir kamel
merceysirkamel

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