dx-5e-2x-4e-x-1-

Question Number 102461 by bemath last updated on 09/Jul/20

\int \frac{d x}{\sqrt{5 e^{2 x} + 4 e^{x} + 1}} = ?

Answered by PRITHWISH SEN 2 last updated on 09/Jul/20

\int \frac{dx}{e^{x} \sqrt{\frac{1}{e^{2 x}} + \frac{4}{e^{x}} + 5}} put \frac{1}{e^{x}} = t \Rightarrow \frac{dx}{e^{x}} = - dt

= - \int \frac{dt}{\sqrt{t^{2} + 4 t + 5}} = - \int \frac{dt}{\sqrt{{(t + 2)}^{2} + 1}}

= - \ln ∣ (t + 2) + \sqrt{t^{2} + 4 t + 5} ∣ + C

= \ln ∣ \frac{e^{x}}{2 e^{x} + 1 + \sqrt{5 e^{2 x} + 4 e^{x} + 1}} ∣ + C

please check .