If f(x)= undersetn arrow ∞ textLim n (x(1/n)-1),(x0) and ∫ xx(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)

Question

If f(x)= undersetn arrow ∞ textLim n (x(1/n)-1),(x>0) and ∫ xx(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

Tardigrade Admin · Accepted Answer

Θ f ( x )= underset n arrow ∞ textLim ( x (1/ n )-1/(1) n )= ln x text and ∫ x x ( ln x +1) dx = x x + c ∴ g ( x )= x text and h ( x )= x ∴ h (1)=1

Q. If $f (x) = n \to \infty Lim n (x^{\frac{1}{n}} - 1), (x > 0)$ 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)

-1

0

1

2

$Θ f (x) = n \to \infty Lim \frac{x ^{\frac{1}{n}} - 1}{\frac{1}{n}} = ln x$
$and \int x^{x} (ln x + 1) d x = x^{x} + c$
$∴ g (x) = x and h (x) = x$
$∴ h (1) = 1$

Q. If f(x)=n→∞Lim​n(xn1​−1),(x>0) and ∫xx(f(x)+1)dx=(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)

-1

0

1

2

Θf(x)=n→∞Lim​n1​xn1​−1​=lnx and ∫xx(lnx+1)dx=xx+c ∴g(x)=x and h(x)=x ∴h(1)=1

Q. If $f (x) = n \to \infty Lim n (x^{\frac{1}{n}} - 1), (x > 0)$ 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)

$Θ f (x) = n \to \infty Lim \frac{x ^{\frac{1}{n}} - 1}{\frac{1}{n}} = ln x$
$and \int x^{x} (ln x + 1) d x = x^{x} + c$
$∴ g (x) = x and h (x) = x$
$∴ h (1) = 1$