## How to integrate Sin(x)/(x)?

Hi guys, I think the question is clear (lol)

How can the Sin(x)/(x) function be integrated? i heared it can be integrated using a series, anyone can explain?

Last added : I remembered the function e^(x^2) how also can it be integrated?

Thanks,

TheDestroyer
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 Recognitions: Gold Member Science Advisor Staff Emeritus Can you think of a series for (sin x)/x?
 Actually I studied taylor series, and using it will not give the general wanted answer as a function, but I heared it can be solved using the fourier series, I don't know, I really completely don't know what to do about it!!!!

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## How to integrate Sin(x)/(x)?

I don't understand your objection: after integrating the Taylor series, you get the Taylor series for the result of the integral.
 Blog Entries: 9 Recognitions: Homework Help Science Advisor According to my ancient Maple,it is a constant times $Si(x)$ + a constant.For some authors (like the ones who produced Maple),the first constant is +1...There are other conventions,though,tipically "normalization" ones,v.Erf(x) ... Daniel.
 Blog Entries: 9 Recognitions: Homework Help Science Advisor THAT was initially $$\int \frac{\sin x}{x} dx$$ Daniel. P.S.Yours can be integrated exactly without any problem...
 Recognitions: Homework Help Science Advisor There are functions created (and used in some circles of mathematics) which basically mean the integrals you are asking: Sine Integral: http://mathworld.wolfram.com/SineIntegral.html Imaginary error function: http://mathworld.wolfram.com/Erfi.html
 Recognitions: Homework Help Science Advisor when you say "integrated" do you mean "antidifferentiated"?
 Blog Entries: 9 Recognitions: Homework Help Science Advisor Of course,what else,he wants to find the antiderivative for those 2 functions... Daniel.
 then there is not antiderivative for them !! even with a series?
 try integration by parts...that is usefull for evaluating an integral composed of two fuctions, in this case, your first integral include the fuctions $$\sin {x}$$ and $$\frac {\11}{x}$$ and your second integral includes $$e^x$$ and $$x^2$$ Remember, integration by parts formula yeilds: $${u}{v} - \int{v}{du}$$ and for the $${e^{x^2}}$$ fuction, when you integrate by parts, make sure you set u equal to the fuction whose derivative will eventually go to zero, otherwise you will have a mess on your hands.
 Blog Entries: 9 Recognitions: Homework Help Science Advisor They have antiderivatives,Si(x) and erf(ix) are (up until mulitplicative and additive constants) their antiderivatives... Daniel.

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 Quote by dagger32 try integration by parts...that is usefull for evaluating an integral composed of two fuctions, in this case, your first integral include the fuctions $$\sin {x}$$ and $$\frac {\11}{x}$$ and your second integral includes $$e^x$$ and $$x^2$$ Remember, integration by parts formula yeilds: $${u}{v} - \int{v}{du}$$ and for the $${e^{x^2}}$$ fuction, when you integrate by parts, make sure you set u equal to the fuction whose derivative will eventually go to zero, otherwise you will have a mess on your hands.
 If you'd like a "better" series than a Taylor series, you might want to know the asymptotic expansion for the function. (By "better", I mean more, faster convergence). $$\int{\frac{\sin{x}}{x}} = -\frac{\cos{x}}{x}-\int{\frac{\cos{x}}{x^2}} = -\frac{\cos{x}}{x}-\frac{\sin{x}}{x^2}-2\int{\frac{\sin{x}}{x^3}}$$ Notice that with each successive integration by parts, the remainder term gets smaller for large values of x. Thus, for all x above a certain value, this series should converge.