1. ## A definite integral

Maria, who has a thing for numbers, was all excited. "Hey, what are you so excited about?" asked Fred. "I just proved that

$\displaystyle \int _{0}^{\pi }\!\sin \left( {x}^{2} \right) {dx}$

is a rational number," said Maria. "No way, that couldn't possibly be a rational number," said Fred. "Not only is it rational but it equals

$\displaystyle {\frac {4616282458010182669}{5974596810169649399}}$

which is a ratio of two prime numbers," said Maria excitedly.

Who is right?
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Note: I only have a solution using Maple. This is a really dumb problem. Please don't waste a lot of time on it.

2. Originally Posted by jbuddenh
$\displaystyle \int _{0}^{\pi }\!\sin \left( {x}^{2} \right) {dx}$
The integral $\displaystyle \int \sin (x^2)~dx$ is known as the Fresnel Integral, and the indefinite integral can't be found in terms of elementary functions.

But it is possible to find the definite integral in some cases.

It's possible to find it when the integration bounds are $\displaystyle (0,\infty)$, $\displaystyle (-\infty,\infty)$, $\displaystyle (-\infty,0)$. Your integration bounds don't fit any of these, so it can't be evaluated.

3. Running your integral through my Voyage 200 I get:

.77265171269. Which is equivalent to your fraction at least to that many places. That's all the places I have.

The solution Maple gives is $\displaystyle \frac{\sqrt{2\pi}}{2}\text{Fresnel}(\sqrt{2\pi})$

4. The numerical approximation of Mathematica shows that your fraction is correct up to 35 digits after decimal (if i didn't count wrong, lol).

5. Thanks to wingless and galactus for their replies. The integral

$\displaystyle \int _{0}^{x }\!\sin \left( {t}^{2} \right) {dt}$

is rational for infinitely many values of x, irrespective of whether the definite integral has a closed form. So far as I know, it is not known whether it is rational or not, for $\displaystyle x=\pi$. In any case Maria's rational number does not equal the desired definite integral. According to Maple the absolute error is about 3.3 x 10^(-39), which is remarkably close.