# What is the arc length of #f(x)= e^(3x) +x^2e^x # on #x in [1,2] #?

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To find the arc length of the function ( f(x) = e^{3x} + x^2e^x ) on the interval ([1, 2]), we need to use the formula for arc length of a function (f(x)) on the interval ([a, b]), which is given by:

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

First, we find the derivative of (f(x)), which is (f'(x)). Then, we plug it into the formula and integrate it over the given interval.

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

[ L = \int_{1}^{2} \sqrt{1 + [3e^{3x} + (2x + x^2)e^x]^2} , dx ]

After integrating this expression, you'll get the arc length of the function (f(x)) on the interval ([1, 2]).

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To find the arc length of the function ( f(x) = e^{3x} + x^2e^x ) on the interval ( x ) in ([1, 2]), we use the formula for arc length:

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

First, we need to find ( f'(x) ), the derivative of ( f(x) ):

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

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

[ L = \int_{1}^{2} \sqrt{1 + (3e^{3x} + 2xe^x + x^2e^x)^2} , dx ]

We evaluate this integral over the interval ([1, 2]) to find the arc length.

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