Question

Obtain: $\int \frac{(e^x-1)}{(e^x-1)\sqrt{e^x+1+e^{-x}}}dx$

Answer 1

$I=\int \frac{e^x-1}{(e^x+1)\sqrt{e^x+1+e^{-x}}}dx$
$\int \frac{e^{\frac{x}{2}}-e^{-\frac{x}{2}}}{(e^{\frac{x}{2}}+e^{-\frac{x}{2}})\sqrt{e^{\frac{x}{2}}+e^{-\frac{x}{2}}}-1}dx$
Substitute $e^{\frac{x}{2}}-e^{-\frac{x}{2}}=t$
Now differentiate w.r.to $t$
$\therefore (e^{\frac{x}{2}}-e^{-\frac{x}{2}})dx=2dt$
$\therefore I=\int \frac{2dt}{t\sqrt{t^2-1}}=2{\sec }^{-1}t=2{\sec }^{-1}(e^{\frac{x}{2}}+e^{-\frac{x}{2}})$
$\therefore I=2{\sec }^{-1}(e^{\frac{x}{2}}-\frac{1}{e^{-\frac{x}{2}}})$
$\therefore I=2{\sec }^{-1}\left(\frac{e^x+1}{\sqrt{e^x}}\right)+c$