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I wondered if there is a way of approximating trigonometric functions in terms of basic functions (possibly trigonometric functions) so that one can derive the indefinite integral of said function. The function I have in mind is the following:

$$f(x)=sin(\frac{sin(2x)}{2}+x)$$

The presented function is the one currently hussling me; however, I also ask for a general method if one exists.

Sidenote:

The mentioned function was intended to be used as an approximation for Jacobian Elliptic Functions with proper constants dependent on k to match the period and "fatness" of the function.

Bruno KM
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  • $\cos\bigg(\dfrac\pi2x\bigg)\simeq\Big(1-x^2\Big)\bigg(1-\dfrac{x^2}{4.5}\bigg)~$ for $~|x|\le1.~$ See here for more information. – Lucian Sep 19 '15 at 20:03

1 Answers1

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You can calculate the taylor-series of the function. In this case, it is difficult to do it by hand. PARI/GP gives : 

? sin(sin(2*x)/2+x)
%1 = 2*x - 2*x^3 + 26/15*x^5 - 376/315*x^7 + 1982/2835*x^9 - 57212/155925*x^11 +
 1075684/6081075*x^13 - 50590052/638512875*x^15 + O(x^17)
Peter
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