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

by TheDestroyer
Tags: integrate, sinx or x
 P: 390 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
 PF Patron Sci Advisor Emeritus P: 16,094 Can you think of a series for (sin x)/x?
 P: 390 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!!!!
PF Patron
Emeritus
P: 16,094

## 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.
 HW Helper Sci Advisor P: 11,722 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.
 HW Helper Sci Advisor P: 11,722 THAT was initially $$\int \frac{\sin x}{x} dx$$ Daniel. P.S.Yours can be integrated exactly without any problem...
 HW Helper Sci Advisor P: 1,123 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
 HW Helper Sci Advisor P: 9,371 when you say "integrated" do you mean "antidifferentiated"?
 HW Helper Sci Advisor P: 11,722 Of course,what else,he wants to find the antiderivative for those 2 functions... Daniel.
 P: 390 then there is not antiderivative for them !! even with a series?
 P: 4 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.
 HW Helper Sci Advisor P: 11,722 They have antiderivatives,Si(x) and erf(ix) are (up until mulitplicative and additive constants) their antiderivatives... Daniel.
HW Helper
 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.
 P: 725 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.