Evaluate-1-e-x-e-3x-e-x-e-x-dx-

Question Number 14399 by tawa tawa last updated on 31/May/17

Evaluate : \int \frac{1 + e^{x} - e^{3 x}}{e^{- x} - e^{x}} dx

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 31/May/17

e^{x} = t \Rightarrow t d x = d t \Rightarrow d x = \frac{d t}{t}

I = \int \frac{1 + t - t^{3}}{\frac{1}{t} - t} . \frac{d t}{t} = \int \frac{t^{3} - t - 1}{t^{2} - 1} d t =

\int \frac{t (t^{2} - 1) - 1}{t^{2} - 1} d t = \int t d t - \int \frac{d t}{t^{2} - 1} =

\frac{1}{2} t^{2} - \frac{1}{2} l n \frac{t - 1}{t + 1} + C = \frac{1}{2} (e^{2 x} - l n \frac{e^{x} - 1}{e^{x} + 1}) + C .

Commented by tawa tawa last updated on 31/May/17

God bless you sir