Can Sin(pi*x^3) Be Integrated Using Elementary Functions?

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Homework Help Overview

The discussion revolves around the integration of the function sin(pi*x^3), which is presented as a final step in a longer problem. Participants are exploring whether this integral can be expressed using elementary functions.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts a substitution method but finds it unhelpful. Some participants suggest that the integral cannot be solved using elementary functions, while others inquire about the possibility of using complex functions or series representations.

Discussion Status

Participants are actively discussing the nature of the integral, with some asserting that it cannot be expressed in terms of elementary functions. There is mention of alternative representations, such as series or the incomplete gamma function, but no consensus on a specific method has been reached.

Contextual Notes

There is an underlying assumption that the techniques typically taught in freshman calculus may not apply to this integral, which raises questions about the limitations of those methods.

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Homework Statement



Need to integrate sin(pi*x^3)

Got to the end of a long question and this is the final step but I can't seem do it!


Homework Equations





The Attempt at a Solution



Tried substitution of u = x^3 and said dx = 1/3x^2 du but this doesn't cancel any x variable. I'm guessing I need to use some trig subsitution or something but don't know which?
 
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You can use all the u-substitutions and trig substitutions you want and you are not going to be able to solve this using elementary functions. This is not integrable in the elementary functions.
 
Are you saying I need to use complex functions?
 
No, it means that there's no way to write the solution in terms of a finite number of usual mathematical operations. You would have to represent the solution as a series or iteration that converges towards the correct result.

The integral of Sin(x^n) can't be written with elementary functions for any n>1. The case n=2 has a special name, the Fresnel integral.
 
I'm saying you can't integrate this using the techniques taught in freshman calculus. This apparently can be integrated using the incomplete gamma function (try your problem on Wolfram Alpha). The incomplete gamma function is not an elementary function.
 

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