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

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To find the arc length of the curve ( f(x) = x - xe^{x^2} ) on the interval ( x \in [2, 4] ), you can use the formula for arc length:

[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

First, find ( \frac{dy}{dx} ) which is the derivative of ( f(x) ):

[ \frac{dy}{dx} = 1 - e^{x^2}(1 + 2x^2) ]

Then, substitute ( \frac{dy}{dx} ) into the arc length formula:

[ L = \int_2^4 \sqrt{1 + \left(1 - e^{x^2}(1 + 2x^2)\right)^2} , dx ]

After evaluating this integral, you'll get the arc length of the curve ( f(x) = x - xe^{x^2} ) on the interval ( x \in [2, 4] ).

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