$u$ substitution doesnt work. I don't see any connection with the Weierstrass substitution either. integration by parts results in a infinite integral series.

  • 1
    $\begingroup$ take a look to the Fresnel Integrals $\endgroup$
    – alexjo
    Commented Dec 19, 2013 at 15:53
  • $\begingroup$ is it possible to express this as a fourier series and than integrate that function? $\endgroup$
    – adam
    Commented Dec 19, 2013 at 16:03
  • $\begingroup$ Fred Rickey has pointed out that the association of the substitution $u=\tan(x/2)$ with Weierstrass is a mistake. It goes back at least to Euler and no one seems to have found any evidence that Weierstrass ever mentioned it. I wonder if the error came only from Stewart's calculus books? $\endgroup$ Commented Dec 19, 2013 at 16:28
  • $\begingroup$ really now? That's something to ponder upon. I just read somewhere this is also a elliptical integral any explanation on that would be greatly appreciated $\endgroup$
    – adam
    Commented Dec 19, 2013 at 19:26

3 Answers 3


this integral does not have a solution in terms of elementary functions. But, you can solve it using series methods. For instance, since

$$ \sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$

$$ \text{then} \; \; \sin (x^2) = \sum \frac{(-1)^n x^{4n+2}}{(2n+1)!}$$

$$ \int \sin (x^2) = \int \sum \frac{(-1)^n x^{4n+2}}{(2n+1)!} = \sum \frac{(-1)^n x^{4n+3}}{(2n+1)!(4n+3)} + K $$

  • $\begingroup$ what is "elementary"? $\endgroup$
    – adam
    Commented Dec 19, 2013 at 15:49
  • $\begingroup$ the well know functions from calculus: the trigonometric, exponentials, etc $\endgroup$
    – ILoveMath
    Commented Dec 19, 2013 at 15:50
  • $\begingroup$ so there are some functions you cannot simply integrate? so how are you supposed to know before d oing so much work that it can't be done? there are nastier functions than this one but yet they turned out ti be integratable. $\endgroup$
    – adam
    Commented Dec 19, 2013 at 15:55
  • 2
    $\begingroup$ just experience will tell you. There is neither magic, nor mistery. Just Experience. $\endgroup$
    – ILoveMath
    Commented Dec 19, 2013 at 16:03
  • $\begingroup$ It can also be proved, see differential Galois theory and this question on MSE or this one $\endgroup$ Commented Dec 19, 2013 at 16:10

Mathematica returns:

$$\sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} x\right)$$

So, unless you consider Fresnel Sine to be an elementary function, that explains your troubles.

  • $\begingroup$ It is not a matter of whether or not you consider a function to be elementary. An elementary function is a well defined concept not subject to opinion. $\endgroup$
    – Jon Claus
    Commented Dec 19, 2013 at 20:02
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    $\begingroup$ @JonClaus Thanks for the terminological enlightenment! As you see from my profile, I am new to this math stuff... $\endgroup$
    – Igor Rivin
    Commented Dec 19, 2013 at 20:27

By Geogebra, the result is:

enter image description here

But if you do it in python with below code:

import sympy as sym
from IPython.display import display

a = Integral(sin(x**2), x)
b = Eq(a,a.doit())

The result is:

enter image description here

But I don't know how to understand the difference.


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