Let (2^1-a+2^1+a),f(a),(3^a+3^-a) be in A.P.

Question

Let $({2}^{1-a}+{2}^{1+a}),f(a),({3}^{a}+{3}^{-a})$ be in A.P. and $\alpha $ be the minimum value of $f(a)$. Then the value of the integral $\int _{{\log }_{e}(\alpha -1)}^{{\log }_{e}(\alpha )}\frac{dx}{({e}^{2x}-{e}^{-2x})}$ is :

Accepted Answer

$\begin{gathered} f(a)=\frac{1}{2}(2({2}^{a}+{2}^{-a})+({3}^{a}+{3}^{-a})) \ a=\frac{1}{2}(2	imes 2+2)=3 \ I={\int }_{\ln 3}^{\ln (3-1)}\frac{{e}^{2x}}{{e}^{4x}-1}dx \ {e}^{2x}=t \ I=\frac{1}{2}{\int }_{9}^{4}\frac{dt}{{t}^{2}-1} \ I=\frac{1}{4}{(\ln (\frac{t-1}{t+1}))}_{9}^{4} \ I=\frac{1}{4}\ln (\frac{4}{3}) \end{gathered}$

Solutions · 1