# what is the$\int \sin (x^2) \, dx$?

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

• take a look to the Fresnel Integrals – alexjo Dec 19 '13 at 15:53
• is it possible to express this as a fourier series and than integrate that function? – adam Dec 19 '13 at 16:03
• 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? – Michael Hardy Dec 19 '13 at 16:28
• 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 – adam Dec 19 '13 at 19:26

## 3 Answers

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

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

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

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

• what is "elementary"? – adam Dec 19 '13 at 15:49
• the well know functions from calculus: the trigonometric, exponentials, etc – ILoveMath Dec 19 '13 at 15:50
• 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. – adam Dec 19 '13 at 15:55
• just experience will tell you. There is neither magic, nor mistery. Just Experience. – ILoveMath Dec 19 '13 at 16:03
• It can also be proved, see differential Galois theory and this question on MSE or this one – Jean-Claude Arbaut Dec 19 '13 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.

• 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. – Jon Claus Dec 19 '13 at 20:02
• @JonClaus Thanks for the terminological enlightenment! As you see from my profile, I am new to this math stuff... – Igor Rivin Dec 19 '13 at 20:27

By Geogebra, the result is:

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())
display(b)


The result is:

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