# Thread: Arc Length of Sine Curve via Integration: square root of sin^2(x/1300) + 1

1. ## Arc Length of Sine Curve via Integration: square root of sin^2(x/1300) + 1

Hey,

So I'm trying to find the arc length of a sine curve.

I need to find the integral of the square root of sin^2(x/1300) + 1

The square root is what screwed it up, cos I don't know how I can go about integrating a trig function with radical.

How can I go about doing this?

Thanks you very much!

2. Originally Posted by AbhiKap55
Hey,

So I'm trying to find the arc length of a sine curve.

I need to find the integral of the square root of sin^2(x/1300) + 1

The square root is what screwed it up, cos I don't know how I can go about integrating a trig function with radical.

How can I go about doing this?

Thanks you very much!
The answer will be in elliptic functions. There is no closed-form solution.

3. Originally Posted by Subhotosh Khan
The answer will be in elliptic functions. There is no closed-form solution.

Thanks for your reply. Can you please elaborate on how I can find the integral of this function if it is an elliptic function.

4. You use a "table of elliptic functions" to write the answer. Again, there is no "closed form solution".
https://en.wikipedia.org/wiki/Elliptic_integral
http://mathworld.wolfram.com/EllipticIntegral.html

5. Originally Posted by HallsofIvy
You use a "table of elliptic functions" to write the answer. Again, there is no "closed form solution".
https://en.wikipedia.org/wiki/Elliptic_integral
http://mathworld.wolfram.com/EllipticIntegral.html
Several functions do not have an elementary antiderivative. Would the square root of the hyperbolic cosh function be one such function?

6. Originally Posted by AbhiKap55
Several functions do not have an elementary antiderivative. Would a hyperbolic cosh function be one such function?
No

d/dx sinh(x) = cosh(x)

7. Originally Posted by Subhotosh Khan
No

d/dx sinh(x) = cosh(x)
My Apologies, Mr. Khan. I was referring to a function f(x) that is equal to the square root of the hyperbolic function cosh.

Would f(x) be un-integratable?

8. Originally Posted by Subhotosh Khan
No

d/dx sinh(x) = cosh(x)
My Apologies, Mr. Khan. I actually meant to say if f(x) is a function equal to the square root of cosh(x)

Would f(x) be un-integratable? Is there a way I can approximate the value?