Question

If $f(x)=\underset{n \rightarrow \infty}{\text{Lim}} n \left(x^{\frac{1}{n}}-1\right),(x>0)$ and $\int x^x(f(x)+1) d x=(g(x))^{h(x)}+c$, where $g(x)$ and $h(x)$ both are linear functions of $x$. If $g(1)=1$, then $h(1)$ is equal to (where $c$ is constant of integration)

• A. -1
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (C)

$ \Theta f ( x )=\underset{ n \rightarrow \infty}{\text{Lim}} \frac{ x ^{\frac{1}{ n }}-1}{\frac{1}{ n }}=\ln x$
$\text { and } \int x ^{ x }(\ln x +1) dx = x ^{ x }+ c$
$\therefore g ( x )= x \text { and } h ( x )= x$
$\therefore h (1)=1$