<s>[INST] Find arc length of curve on the given interval I was asked to find the arc length of the curve of the following curve:
$24xy = x^4 + 48$ from $x = 2$ to $x = 4$
This has turned out to be a very difficult problem, I get stuck using the arc length formula with the derivative I have calculated.
 [/INST] We can put
$$\begin{cases} x&=t \\y&=\frac{1}{24}t^3+2t^{-1}\end{cases}$$
Arc length formula $$s=\int_{2}^{4}{\sqrt{(x')^2+(y')^2}\text{d}t}$$ give us 
\begin{align*}
s&=\int_{2}^{4}{\sqrt{1+\left(\frac{1}{8}t^2-2t^{-2}\right)^2}\text{d}t}\\
 &=\int_{2}^{4}{\sqrt{\left(\frac{1}{8}t^2+2t^{-2}\right)^2}\text{d}t}\\
 &=\int_{2}^{4}{\left(\frac{1}{8}t^2+2t^{-2}\right)\text{d}t}\\
 &=\left[\frac{1}{24}t^3-2t^{-1}\right]_2^4\\
 &=\frac{17}{6}
\end{align*}
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