Question

The indefinite integral $\int e^{e^{x} \frac{x e^{x} \cdot \ln x+1}{x}} d x$ simplifies to (where, $c$ is the constant of integration)

• A. $x \ln (\ln x)+c$
• B. $e^{e^{x}}+c$
• C. $\frac{e^{e^{x}}}{x}+C$
• D. $e^{e^{x}} \cdot \ln x+c$

Answer 1

Answer: (D)

Given integral is $I=\int e^{e^{x}} \cdot e^{x} \cdot \ln x+\frac{e^{e^{x}}}{x} d x$
Let, $e^{e^{x}}=f x$ and $\ln x=g x$
$\therefore I=\int f^{\prime} x g x+f x g^{\prime} x d x$
$=f x \cdot g x+c$
$=e^{e^{x}} \cdot \ln x+c$