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What is the integral of sinx2?

Updated: 4/28/2022
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15y ago

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If you mean the the integral of sin(x2)dx, It can only be represented as an infinite series or a unique set of calculus functions known as the Fresnel Integrals (Pronounced Frenel). These functions, S(x) and C(x) are the integrals of sin(x2) ans cos(x2) respectively. These two integrals have some interesting properties. To find out more, go to:

http://en.wikipedia.org/wiki/Fresnel_integral

I hope this answers your question.

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What is the integral of sin x squared?

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 Thank you for posing this question.


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