The-solution-of-the-equation-log-2-x-1-e-x-1-dx-pi-6-is-given-by-

Question Number 53697 by gunawan last updated on 25/Jan/19

The solution of the equation

\underset{\log 2}{\int^{x}} \frac{1}{\sqrt{e^{x} - 1}} d x = \frac{π}{6} is given by

Commented by Abdo msup. last updated on 25/Jan/19

l e t A (x) = \int_{l o g 2}^{x} \frac{d t}{\sqrt{e^{t} - 1}} \Rightarrow A (x) =_{e^{t} = u} \int_{2}^{e^{x}} \frac{d u}{u \sqrt{u - 1}}

=_{\sqrt{u - 1} = α} \int_{1}^{\sqrt{e^{x} - 1}} \frac{2 α d α}{(1 + α^{2}) α} = 2 \int_{1}^{\sqrt{e^{x} - 1}} \frac{d α}{1 + α^{2}}

= 2 {[a r c t a n (α)]}_{1}^{\sqrt{e^{x} - 1}} = 2 {a r c t a n (\sqrt{e^{x} - 1}) - \frac{π}{4}}

= 2 a r c t a n (\sqrt{e^{x} - 1}) - \frac{π}{2} s o

A (x) = \frac{π}{6} \Leftrightarrow 2 a r c t a n (\sqrt{e^{x} - 1}) = \frac{π}{2} + \frac{π}{6} \Rightarrow

2 a r c t a n (\sqrt{e^{x} - 1}) = \frac{4 π}{6} = \frac{2 π}{3} \Rightarrow

a r t a n (\sqrt{e^{x} - 1}) = \frac{π}{3} \Rightarrow \sqrt{e^{x} - 1} = t a n (\frac{π}{3}) \Rightarrow

\sqrt{e^{x} - 1} = \sqrt{3} \Rightarrow e^{x} - 1 = 3 \Rightarrow e^{x} = 4 \Rightarrow x = 2 l n (2) .