Input
- x0, a real number greater than 0
Output
- I, a numerical estimate of the integral
x0
/
I = | cosh(x) / sqrt(cosh(x0) - cosh(x)) dx
/
0Example:
x0=1.0 --> I = 2.6405789412796
Remarks:
- Aim at a relative precision better than 1e-10
- The problem arises studying the frictionless movement of a mass point on a hanging wire, which follows the curve cosh(x).
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Please, add to the problem description that for values smaller than one, we should use 1 + x^2/2 for approximating the integrand. For instance, the answer provided by vpaintegral (Symbolic Math Toolbox; commented in my solution) is not accepted by the 4th test case: I believe that vpaintegral returns the exact solution*. The precision for the 4th test should be way smaller, 1e-2/1e-3, or the problem description needs to be changed.
* vpaintegral can handle values that cause the MATLAB integral function to overflow or underflow.
As a sanity check, I even did Monte Carlo Integration for the 4th case, and it still did not work, because the problem is not asking for the exact solution, but an approximation of an approximation (and that's why the requested precision should be smaller).
What is the point of this question? Everyone is just using built-in integral functions anyway. You cannot actually code a solution to this problem because the precision is way too high.