Question

If $f\left(x\right)=\sqrt{x},g\left(x\right)=e^x-1,$ and $\int fog\left(x\right)dx=Afog\left(x\right)+Bta{n}^{-1}\left(fog\right(x\left)\right)+C$, then $A+B$ is equal to

Answer 1

$fog\left(x\right)=\sqrt{e^x-1}$
$\therefore I=\int \sqrt{e^x-1}dx$
Putting $\sqrt{e^x-1}=t\Rightarrow dt=\frac{e^x}{2\sqrt{e^x-1}}dx$
$\Rightarrow I=\int \frac{2t^2}{t^2+1}dt=2\int dt-2\int \frac{1}{1+t^2}dt$
$=2t-2{\tan }^{-1}t+C$
$=2\sqrt{e^x-1}-2{\tan }^{-1}\left(\sqrt{e^x-1}\right)+C$
$=2fog\left(x\right)-2{\tan }^{-1}\left(fog\right(x\left)\right)+C$
$\therefore A+B=2+(-2)=0$