# How do you find the arc length of the curve #y= ln(sin(x)+2)# over the interval [1,5]?

You'll need numerical methods.

You need to find

I can't find a closed form antiderivative (Wolfram Alpha gives one that uses elliptic integrals of the first and third kind. https://tutor.hix.ai )

Use your favorite approximation technique to get an answer close to 4.23701 (also from WolframAlpha).

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To find the arc length of the curve (y = \ln(\sin(x) + 2)) over the interval ([1, 5]), you can use the arc length formula:

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

where (a = 1), (b = 5), and (\frac{dy}{dx}) is the derivative of (y) with respect to (x).

First, find (\frac{dy}{dx}) by differentiating (y) with respect to (x). Then, substitute it into the arc length formula and integrate over the interval ([1, 5]).

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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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