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I'm not looking for the solved integrals, I'm just looking for a simple explanation for why

$\displaystyle\int\cos(x^2)dx$

is so much more complicated than

$\displaystyle\int\cos(x)dx$

With simple I mean something I can say to explain the difference to a person just starting to learn about integration. Perhaps something visual?

Mattis
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  • I can't really think of a "simple" explanation; the rigorous route would be to use Risch to show that the Fresnel integral is nonelementary... – J. M. ain't a mathematician May 31 '13 at 10:22
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    Before you designate something as more or less complicated, you need to be precise about what complicated means. In which sense is $\int \cos(x)dx$ simpler than $\int \cos(x^2)dx$? – Ittay Weiss May 31 '13 at 10:26
  • @IttayWeiss You are right, I added more info. – Mattis May 31 '13 at 10:34
  • @Mattis I was not referring to the word 'simple' as the one needing further explanation, but rather to the word 'complicated' in the second line of text. In which sense is the first integral more complicated than the second one? – Ittay Weiss May 31 '13 at 10:36
  • @IttayWeiss Ah! Well, a beginner would try and get to sin(y^3)/3 or something like that. But you can't, which to me is more complicated. – Mattis May 31 '13 at 10:59

1 Answers1

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For one thing, look at their graphs. Since $x$ increases at a constant rate, $\cos(x)$ has a constant period but $x^2$ increases faster and faster for larger $x$ so the "period" of $\cos(x^2)$ keeps getting smaller and smaller.

Michael Albanese
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john
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