Integral of (cos(2x))^0.5

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I tried to do a arc length integral s, for y as an elliptic function of x. But as I continued with the integration, I found myself at the above integral(cosine of 2x). I quickly substituted cos(2x) with A and carried on but got stuck after about a step or two. The new problem now became (A^0.5)/(1-A^2)^0.5
I tried integration by parts, a lot of substitutions, and nothing worked. Then I thought I should give this to wolframalpha. It gave the results but in, what I only recently found out, elliptic functions. So, does this mean there are no closed solutions/expressions to integrals like these? And does that mean there is no exact formula for applied mathematical problems involving these, like here the perimeter of an ellipse?
 

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  • #2
No. For these type of problems, and in reality in physics especially, closed-form solutions are extremely rare!

Indeed, the integral as you posted it only has the elliptical function as a solution, which is a "special function" solution.
There are others for other such problems for example, such as the hypergeometric function, Laguerre polynomials, etc.

Although, there is no closed-form solution, you can perform numerical integrations. Since Wolfram Alpha is based on Mathematica, research the function NIntegrate to see how this works.

For example, to evaluate your integral from 0 to 1, you can try:
NIntegrate[Sqrt[Cos[2x]],{x,0,1}],
and experiment with that.
 

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