Calculating Integral of sin(x^2): A Beginner's Guide

[Gina88]

How can I calculate the integral of sin(x^2) from zero to infinity? Do I need to use the residue theorem?
I need a detailed answer, because I'm very new in this subject.
I don't even know where to start.
Thanks for any kind of help :-)

 

[CompuChip]

I solved it, but I don't know what your level of calculus is so if this is too hard.

First I extended the integral to -infinity to infinity and wrote it as*
$$\int \sin(x^2) \, \mathrm dx = \operatorname{Im} \int e^{i x^2} \, \mathrm dx$$.
I happen to know that
$$\int e^{- a x^2} \, \mathrm dx = \sqrt{ \frac{\pi}{a} }$$
if Re(a) > 0, and that the limit Re(a) -> 0 is well-defined.

Combining all that allowed me to solve the integral, eventually I got
$$\frac12 \sqrt{\frac{\pi}{2}}$$.

* All integrations are over $]-\infty, \infty[$ unless otherwise specified[/size]

 

[Gina88]

Okay, thanks.
I've got 2 questions: Any chance I can solve this problom with the residue theorem?
I know that i can write sin(x^2) like this:x^2 - x^6/3! + x^10/5! - x^14/7!...
But then what?
And i can't see clearly enough the text you wrote. Do I need some kind of program in order to view your solution?

 

[Cyosis]

Yes you can use the residue theorem to solve this integral. Use the contour shown in http://en.wikipedia.org/wiki/Fresnel_integral#Error_function.

Hint: start with the function $e^{iz^2}$ and evaluate it along the different parts of the contour.

[quote='Gina88]And i can't see clearly enough the text you wrote. Do I need some kind of program in order to view your solution? [/quote]

Are you using internet explorer by any chance, perhaps an outdated version? If so installing Firefox should solve the display issues.