# What is the arclength of #f(x)=x^2e^(1/x)# on #x in [1,2]#?

We are using the Formula

This integral can not expressed by the known elementary functions.

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To find the arc length of the function ( f(x) = x^2e^{1/x} ) on the interval ([1,2]), we use the formula for arc length, which is given by:

[ L = \int_{a}^{b} \sqrt{1 + \left( f'(x) \right)^2} , dx ]

Where ( f'(x) ) represents the derivative of the function ( f(x) ).

First, we need to find the derivative of ( f(x) ), which involves using the product rule and the chain rule. After finding ( f'(x) ), we substitute it into the arc length formula.

The derivative of ( f(x) ) is given by:

[ f'(x) = 2xe^{1/x} - \frac{1}{x^2}e^{1/x} ]

Now, we substitute ( f'(x) ) into the arc length formula:

[ L = \int_{1}^{2} \sqrt{1 + \left( 2xe^{1/x} - \frac{1}{x^2}e^{1/x} \right)^2} , dx ]

After substituting, we integrate the expression from ( x = 1 ) to ( x = 2 ) to find the arc length ( L ). This integral may be evaluated numerically using numerical integration techniques or appropriate software.

Once the integral is evaluated, the resulting value will be the arc length of the function ( f(x) = x^2e^{1/x} ) on the interval ([1,2]).

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