Which of the following integrals gives the length of the curve \(y=\ln x\) from \(x=1\) to \(x=2\) ?

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The arc length for a curve is given by the equation:

$$ L = \int\limits_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}   dx $$

Therefore,

$$ f(x)=\ln x \hskip{2em} f'(x)=\frac{1}{x} $$ $$ \displaystyle L = \int\limits_1^2 \sqrt{1+ \left(\frac{1}{x}\right)^2}   dx $$ $$ \displaystyle L = \boxed{\int\limits_1^2 \sqrt{1+\frac{1}{x^2}}   dx} $$