# What is the arc length of #f(x)=xe^(2x-3) # on #x in [3,4] #?

Arc length is given by:

Rearrange:

Hence

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To find the arc length of the function ( f(x) = xe^{2x-3} ) on the interval ([3,4]), we use the arc length formula:

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

First, find the derivative of ( f(x) ):

[ f'(x) = e^{2x-3} + x(e^{2x-3})(2) ]

Now, compute ( (f'(x))^2 ):

[ (f'(x))^2 = (e^{2x-3})^2 + (2xe^{2x-3})^2 ]

[ (f'(x))^2 = e^{4x-6} + 4x^2e^{4x-6} ]

Now, plug ( (f'(x))^2 ) into the arc length formula and integrate over the interval ([3,4]):

[ L = \int_{3}^{4} \sqrt{1 + e^{4x-6} + 4x^2e^{4x-6}} , dx ]

This integral represents the arc length of ( f(x) = xe^{2x-3} ) on the interval ([3,4]). You can solve this integral using numerical methods or software tools.

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