advanced-calculus-prove-that-I-1-t-4-6t-2-1-ln-ln-t-1-t-2-3-dt-2G-pi-G-catalan-constant-

Question Number 125462 by mnjuly1970 last updated on 11/Dec/20

... ♣advanced calculus♣... ⧫⧫ prove that: I=∫_1 ^( ∞) (((t^4 −6t^2 +1)ln(ln(t)))/((1+t^2 )^3 ))dt=((2G)/π) G : catalan constant...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:…\:\clubsuit{advanced}\:\:{calculus}\clubsuit… \\ $$$$\:\:\:\blacklozenge\blacklozenge\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\mathrm{I}=\int_{\mathrm{1}} ^{\:\infty} \frac{\left({t}^{\mathrm{4}} −\mathrm{6}{t}^{\mathrm{2}} +\mathrm{1}\right){ln}\left({ln}\left({t}\right)\right)}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }{dt}=\frac{\mathrm{2G}}{\pi} \\ $$$$\:\:\mathrm{G}\::\:\:{catalan}\:\:{constant}… \\ $$

Answered by mindispower last updated on 13/Dec/20