<s>[INST] Find the exact length of the curve $y=\frac 12 x^2- \frac 12 \ln(x)$ 
Find the exact length of the curve $y = \frac 12 x^2- \frac 12 \ln(x)$, for $2 \le x \le 4$.

My attempt:
\begin{align}
L&= \int_2^4 \sqrt{1+\left[x-\frac 1{2x} \right]^2} \, dx \\
&= \int_2^4 \sqrt{1+x^2+\frac 1{4x^2}-1} \, dx \\
&= \int_2^4 \sqrt{x^2+\frac 1{4x^2}} \, dx
\end{align}
But I am stuck here. How can I proceed from this step onwards?
 [/INST] It is possible that there is a typo in the question: often in arclength problems, the constants are chosen so that $\sqrt{1+\left(\frac{dy}{dx}\right)^2}$ miraculously turns out to be the square of something nice.  However, let us assume there is no typo.
We want to integrate $\frac{1}{2x}\sqrt{4x^4+1}$. Multiply top and bottom by $x^3$,  and make the change of variable $4x^4+1=u^2$. We arrive at a standard integral. 
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