Question

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

• A. ${\cos }^{-1}\left(\frac{e^x+2}{3}\right)+c$
• B. ${\cos }^{-1}\left(\frac{e^x-3}{2}\right)+c$
• C. ${\sin }^{-1}\left(\frac{e^x+2}{3}\right)+c$
• D. ${\sin }^{-1}\left(\frac{e^x-3}{2}\right)+c$

Answer 1

The correct option is A ${\cos }^{-1}\left(\frac{e^x+2}{3}\right)+c$

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

Putting $e^x=t$

$e^{x}dx=dt$

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

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

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

$=\ln |(t-2)+\sqrt{{(t-2)}^2+1}|+c$

$=\ln |(e^x-2)+\sqrt{e^{2x}-4e^x+5}|+c$