Let (21−a+21+a),f(a),(3a+3−a)(2^{1-a}+2^{1+a}) , f(a), (3^{a}+3^{-a})(21−a+21+a),f(a),(3a+3−a) be a A. P. and α be the minimum value of f(a)f(a)f(a). Then the value of the integral ∫loge(α−1)loge(α) dxe2x−e−2x\int\limits_{\log_{e}(\alpha-1)}^{\log_{e}(\alpha)} \;\dfrac{dx}{e^{2x} - e^{-2x}}loge(α−1)∫loge(α)e2x−e−2xdx is :
12loge( 43)\;\dfrac{1}{2}\log_{e}\left(\;\dfrac{4}{3}\right)21loge(34)
14loge( 43)\;\dfrac{1}{4}\log_{e}\left(\;\dfrac{4}{3}\right)41loge(34)
12loge( 85)\;\dfrac{1}{2}\log_{e}\left(\;\dfrac{8}{5}\right)21loge(58)
14loge( 85)\;\dfrac{1}{4}\log_{e}\left(\;\dfrac{8}{5}\right)41loge(58)