Show-that-1-1-dx-5-cosh-x-13-sinh-x-1-2-log-e-15e-10-3e-2-

Question Number 29105 by tawa tawa last updated on 04/Feb/18

Show that : \int_{- 1}^{1} \frac{dx}{5 \cosh (x) + 13 \sinh (x)} = \frac{1}{2} \log_{e} (\frac{15 e - 10}{3 e + 2})

Commented by abdo imad last updated on 04/Feb/18

I = \int_{- 1}^{1} \frac{d x}{5 \frac{e^{x} + e^{- x}}{2} + 13 \frac{e^{x} - e^{- x}}{2}}

= \int_{- 1}^{1} \frac{2 d x}{18 e^{x} - 8 e^{- x}} t h e c h . e^{x} = t g i v e

I = \int_{e^{- 1}}^{e} \frac{2}{18 t - \frac{8}{t}} \frac{d t}{t} = \int_{e^{- 1}}^{e} \frac{2}{18 t^{2} - 8} d t

I = \int_{e^{- 1}}^{e} \frac{d t}{9 t^{2} - 4} => 9 I = \frac{3}{4} \int_{e^{- 1}}^{e} (\frac{1}{t - \frac{2}{3}} - \frac{1}{t + \frac{2}{3}}) d t

I = \frac{1}{12} l n ∣ \frac{t - \frac{2}{3}}{t + \frac{2}{3}} ∣_{e^{- 1}}^{e} = \frac{1}{12} (l n ∣ \frac{3 e - 2}{3 e + 2} ∣ - l n ∣ \frac{3 e^{- 1} - 2}{3 e^{- 1} + 2} ∣ \dots .

Commented by tawa tawa last updated on 04/Feb/18

God bless you sir .