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

[SteveDC]

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?

 

[D H]

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.

 

[SteveDC]

Are you saying I need to use complex functions?

 

[hilbert2]

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.

 

[D H]

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.