5D%7Bx%7D%20%20%5Ctan%5E%7B-%201%7D%20%28x%29%7D%7B%281%20%2B%20%20%7Bx%7D%5E%7B%20%5Cphi%7D%20%7B%29%7D%5E%7B2%7D%20%20%7D%20%7B%7D%5E%7B%7D%20%20%20%7B%7D%20%20%20%5C%3A%20dx%5C%5C%20" id="TexFormula1" title=" \rm \int_{0}^ \infty \frac{ \sqrt[ \scriptsize\phi]{x} \tan^{- 1} (x)}{(1 + {x}^{ \phi} {)}^{2} } {}^{} {} \: dx\\ " alt=" \rm \int_{0}^ \infty \frac{ \sqrt[ \scriptsize\phi]{x} \tan^{- 1} (x)}{(1 + {x}^{ \phi} {)}^{2} } {}^{} {} \: dx\\ " align="absmiddle" class="latex-formula">
1 answer:
With ϕ ≈ 1.61803 the golden ratio, we have 1/ϕ = ϕ - 1, so that
Replace :
Split the integral at x = 1. For the integral over [1, ∞), substitute :
The integrals involving tan⁻¹ disappear, and we're left with
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