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Question Number 186800 by universe last updated on 10/Feb/23
show that  ∫_0 ^( ∞ ) ((tan^(−1) 𝛂x tan^(−1) 𝛃x)/x^2 )dx = (𝛑/2)log{(((𝛂+𝛃)^(𝛂+𝛃) )/(𝛂^𝛂 𝛃^𝛃 ))}
$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\int_{\mathrm{0}} ^{\:\infty\:} \frac{\boldsymbol{\mathrm{tan}}^{−\mathrm{1}} \boldsymbol{\alpha{x}}\:\boldsymbol{\mathrm{tan}}^{−\mathrm{1}} \boldsymbol{\beta{x}}}{\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}\:=\:\frac{\boldsymbol{\pi}}{\mathrm{2}}\mathrm{log}\left\{\frac{\left(\boldsymbol{\alpha}+\boldsymbol{\beta}\right)^{\boldsymbol{\alpha}+\boldsymbol{\beta}} }{\boldsymbol{\alpha}^{\boldsymbol{\alpha}} \boldsymbol{\beta}^{\boldsymbol{\beta}} }\right\}\: \\ $$

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