Let f(x) = intlimits_1^x {left( {tln(t) - fra | vedclass

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(D) Given $f(x) = \int\limits_1^x {\left( {t\ln(t) - \frac{{\ln(t)}}{t}} \right)dt}$.Using the Fundamental Theorem of Calculus,$f'(x) = x\ln(x) - \fra

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$f(x)$ is monotonic.

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(D) Given $f(x) = \int\limits_1^x {\left( {t\ln(t) - \frac{{\ln(t)}}{t}} \right)dt}$.Using the Fundamental Theorem of Calculus,$f'(x) = x\ln(x) - \frac{{\ln(x)}}{x}$.Factoring the expression,$f'(x) = \ln(x) \left( x - \frac{1}{x} \right) = \ln(x) \left( \frac{x^2 - 1}{x} \right) = \frac{{\ln(x)(x - 1)(x + 1)}}{x}$.For $x > 1$,we know that $\ln(x) > 0$,$(x - 1) > 0$,$(x + 1) > 0$,and $x > 0$.Therefore,$f'(x) > 0$ for all $x > 1$.Since the derivative $f'(x)$ is strictly positive for all $x$ in the domain $(1, \infty)$,the function $f(x)$ is strictly increasing.Thus,$f(x)$ is monotonic.

Let $f(x) = \int\limits_1^x {\left( {t\ln(t) - \frac{{\ln(t)}}{t}} \right)dt}$ for $x > 1$. Then:

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