How do I integrate sin(x^2) please? Not (sin(x))^2.

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Discussion Overview

The discussion revolves around the integration of the function sin(x^2), with participants clarifying that it is distinct from (sin(x))^2. The scope includes mathematical reasoning and the challenges associated with integrating non-elementary functions.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests using the equality sin(u) = (e^(iu) - e^(-iu)) / (2i) to approach the integral, assuming familiarity with erf(x) and erfi(x).
  • Another participant states that the integral cannot be expressed in terms of elementary functions.
  • Some participants note that many integrals cannot be solved in terms of elementary functions, indicating a broader context of understanding integration challenges.
  • A participant asserts that while sin(x^2) is continuous and thus integrable, it is not integrable in terms of elementary functions.
  • There is a contention regarding the ease of integration, with one participant arguing that difficulty should not affect the classification of integrability.

Areas of Agreement / Disagreement

Participants generally agree that sin(x^2) cannot be integrated in terms of elementary functions, but there is disagreement about the implications of this fact and the nature of integrability itself.

Contextual Notes

Some statements reflect assumptions about the definitions of integrability and the nature of functions, which may not be universally accepted. The discussion also highlights the complexity of certain integrals without resolving the underlying mathematical nuances.

esisk
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How do I integrate sin(x^2) please?

Not (sin(x))^2. Thanks again
 
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Use a certain equality of the sine function, namely...

sin(u) = \frac{e^{iu}-e^{-iu}}{2i}

Replace u with x^2 and you're good to go, assuming you know erf(x) and erfi(x).
 


In other words, the integral cannot be done in terms of elementary functions.
 


HallsofIvy said:
In other words, the integral cannot be done in terms of elementary functions.

But that's true for most integrals. I used to freak out when I couldn't solve an integral in terms of elementary functions, but I know better now.
 


Char. Limit said:
But that's true for most integrals. I used to freak out when I couldn't solve an integral in terms of elementary functions, but I know better now.
Actually, it is true for "almost all" functions!:wink:
 


esisk said:
How do I integrate sin(x^2) please?

Not (sin(x))^2. Thanks again

hey this function is not integerable
 


esisk said:
How do I integrate sin(x^2) please?

Not (sin(x))^2. Thanks again

tauchatri said:
hey this function is not integerable
Non-sense. It is continuous, therefore it is integrable. It is simply not integrable in terms of elementary functions.
 


HallsofIvy said:
Non-sense. It is continuous, therefore it is integrable. It is simply not integrable in terms of elementary functions.

i only mean that you duffer.it's not easy to integrate
 
  • #10


tauchatri said:
i only mean that you duffer.it's not easy to integrate

Difficulty really shouldn't matter with integration, I mean, this function isn't easy to integrate either:

\frac{sin(x^2) tan(x^3)}{x}+3 x^2 log(x) sin(x^2) sec^2(x^3)+2 x log(x) cos(x^2) tan(x^3)

But we can still express the integral in terms of elementary functions.
 

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