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The integral of sin x2 is one of the Fresnel Integrals.
It does not have a closed form solution. However, you can calculate a series solution by integrating the Taylor series, as follows:
The Taylor series expansion about x = 0 for sin x is
sin x = x - (x3/3!) + (x5/5!) - (x7/7!) +/- ...
Substitution of x2 for x yields
sinx2 = x2 - (x6/3!) + (x10/5!) - (x14/7!) +/- ...
Term-wise integration, using the power rule gives
{integral}sinx2 = (x3/3) - (x7/7*3!) + (x11/11*5!) - (x15/15*7!) +/- ...
This is the answer. It is the Fresnel Integral S(x).
There is a similar one for the integral of cos x2, called C(x).
It can be written in more compact form:
S(x) = (Sum from n = 1 to infinity) of (-1)n x4n+3/(4n+3)*(2n+1)!
It looks better in Sigma notation, with fractions, but
if you work out the first 4 terms, you will see agreement with the result for integrating the series expansion.
Here is a link to Fresnel Integral on Wikipedia:
http://en.wikipedia.org/wiki/Fresnel_integral
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