Question

Evaluate:
$\int \frac{e^x}{\sqrt{5-4e^x-e^{2x}}}dx$

Answer 1

$I=\int \frac{e^x}{\sqrt{5-4e^x-e^{2x}}}dx$

Now, put $e^x=t⟹e^{x}dx=dt$

Therefore,

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

$I=\int \frac{dt}{\sqrt{-\left[\right(t+2{)}^2-5-4]}}$

$I=\int \frac{dt}{\sqrt{9-(t+2{)}^2}}=\int \frac{dt}{\sqrt{(3{)}^2-(t+2{)}^2}}$

$I={\sin }^{-1}\left(\frac{t+2}{3}\right)+C$

$I={\sin }^{-1}\left(\frac{e^x+2}{3}\right)+C$