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

And so the required Arc Length is given by:

Using Wolfram Alpha this integral evaluates to:

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To find the arc length of ( f(x) = x \ln x ) on the interval ([1, e^2]), you need to integrate the square root of (1 + (f'(x))^2) over the given interval, where (f'(x)) is the derivative of (f(x)).

First, find the derivative of ( f(x) ): [ f'(x) = 1 + \ln x ]

Then, calculate the square root of (1 + (f'(x))^2): [ \sqrt{1 + (1 + \ln x)^2} ]

Now, integrate this expression with respect to (x) over the interval ([1, e^2]):

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

This integral represents the arc length of ( f(x) = x \ln x ) on the interval ([1, e^2]). You can solve it using numerical methods or integration techniques.

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To find the arc length of the curve ( f(x) = x \ln(x) ) on the interval ([1, e^2]), you can use the formula for arc length:

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

First, find the derivative of ( f(x) ) which is ( f'(x) ). Then substitute ( f'(x) ) into the formula and integrate it over the given interval. This will give you the arc length of the curve on the specified interval.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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