Question

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

• A. $xln \left(ln ⁡ x\right) + c$
• 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 $I=\int\left(e^{e^{x}} \cdot e^{x} \cdot \ln x+\frac{e^{e^{x}}}{x}\right) d x$
Let, $e^{e^{x}}=f(x)$ and $\ln x=g(x)$
$\therefore I=\int\left(f^{\prime}(x) g(x)+f(x) g^{\prime}(x)\right) d x$
$=f(x) \cdot g(x)+c$
$=e^{e^{x}} \cdot \ln x+c$