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Is there any way of integrating this trigonometric function $$\int \cos(x^2)dx$$ ? Wolfram alpha straight away gives this $$\sqrt{\frac{\pi }{2}}C\left ( \sqrt{\frac{2}{\pi }} x\right )+\text{constant}$$ without showing any steps.

It would be very helpful if someone could show the steps for me please..

Thank You

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There are no steps to show -- $\int \cos(x^2)dx$ is the definition of the Fresnel integral $C(x)$. The $\sqrt{\pi/2}$ factors must be due to differing scaling conventions. –  Henning Makholm Oct 30 '11 at 9:01
    
@Henning: Yes indeed; the Fresnel integrals find use in optics (as part of the definition of the Cornu spiral among other things), and the scaling convention chosen simplifies the form of the applicable optical formulae... –  J. M. Oct 30 '11 at 9:20

2 Answers 2

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Proposed answer removed by author due to remarks below and to preserve the comments. Thanks for your comments. I should use glasses.

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I think you're confusing $\sin(x^2)$ and $(\sin x)^2$. –  Gerry Myerson Oct 30 '11 at 9:44
    
You have it wrong; what you put in there is another Fresnel integral (the one that is usually paired with the OP's Fresnel integral). –  J. M. Oct 30 '11 at 9:47
    
Please tell me the difference between sin(x^2) and (sinx)^2 –  alok Nov 20 '11 at 6:27
    
sin(x^2) = sin(x*x) - However, (sinx)^2 = sin(x)*Sin(x) - The two curves are totally different. When you have a function you can't distribute it unless it supports this feature. So it is NOT generally true to say that f(x*x)=f(x).f(x) - One function that supports this is f(x)=1. –  Emmad Kareem Nov 20 '11 at 8:48

This integral $\int{\cos{(x^2)}dx}$ cannot be expressed by so called elementary functions. Here is a link about similar integrals: http://www.math.unt.edu/integration_bee/AwfulTruth.html

Sincerely,

Tigran

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