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

• A. 1
• B. 2
• C. 3

Answer 1

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