Question

The value of $\displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x} \left(1 + e^{2 \sqrt{x}}\right)}dx$ is equal to (where, $C$ is the constant of integration)

• A. $\left(tan\right)^{- 1} \left(2 e^{\sqrt{x}}\right)+C$
• B. $ln \left(\frac{1 + e^{\sqrt{x}}}{1 - e^{\sqrt{x}}}\right)+C$
• C. $2\left(tan\right)^{- 1} \left(e^{\sqrt{x}}\right)+C$
• D. $\left(\left(tan\right)^{- 1} x\right)e^{\sqrt{x}}+C$

Answer 1

Answer: (C)

Let, $I=\int \frac{e^{\sqrt{x}}}{\sqrt{x}\left(1+e^{2} \sqrt{x}\right)} d x$
Substituting $e^{\sqrt{x}}=t$ $\frac{e^{\sqrt{x}}}{2 \sqrt{x}} d x=d t$
$I=2 \int \frac{d t}{1+t^{2}}$
$=2\left(\tan ^{-1} t\right)+C$
$I=2 \tan ^{-1}\left(e^{\sqrt{x}}\right)+C$