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

Solution 1:

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

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

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

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

Solution 2:

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.

Solution 3:

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.