# Integral of sin(x^2) help

1. Jul 29, 2009

### unseenoi

integral of sin(x^2)

2. Jul 29, 2009

### Cyosis

Are you familiar with complex integration and the residue theorem?

3. Jul 29, 2009

### unseenoi

no i am not

4. Jul 29, 2009

### Hootenanny

Staff Emeritus
In that case, I would try a substitution.

5. Jul 29, 2009

### Cyosis

Seeing as you say you're getting ready for school, are you still in high school? Also I just noticed that you didn't specify an interval to integrate over. Did you just make up this integral yourself? The reason I am asking this is that this function does not have a primitive function in terms of elementary functions.

6. Jul 29, 2009

### Hootenanny

Staff Emeritus
Well noted Cyosis, I presumed that by 'school' the OP meant grad school, which looking back now may have not been a wise assumption.

7. Jul 29, 2009

### g_edgar

$\int \sin(x^2)\,dx$ is not elementary.

So "hints" like "try substitution" are not helpful.

8. Jul 29, 2009

### Hootenanny

Staff Emeritus
Indeed it is, as has already been pointed out.
Really? How about substituting u=x2, then expanding sin(u) about u=0 and performing term-wise integration? Does this not give the power-series definition of the Fresnel function S(x)?

9. Jul 29, 2009

### n!kofeyn

I think your substitution hint implied either u-substitution or integration by parts. There is no need to make a substitution to expand sin(x2) out into its power series.

10. Oct 28, 2009

### Barkan

it is a fresnel intetgral

11. Oct 28, 2009

### Barkan

There is no systematic way to compute Fresnel integrals as I know.
But there are several approximation methods

I found Peter L. Volegov's code in Matlab central. It uses a method proposed in the following : (ith an error of less then 1x10-9)

Klaus D. Mielenz, Computation of Fresnel Integrals. II
J. Res. Natl. Inst. Stand. Technol. 105, 589 (2000), pp 589-590

Or simply wiki Fresnel Integrals

12. Oct 28, 2009

### Barkan

by the way it is suprising that nobody above heard of Fresnels.

13. Oct 28, 2009

### Count Iblis

It is useful if you want to derive an asymptotic expression for the case of the integral from zero to R for large R.

14. Oct 28, 2009

### Count Iblis

If you're integrating from 0 to R, then for small R, you simply integrate the Taylor expansion term by term.

If R is large, you write the integral as an integral from zero to infinity minus the integral from R to infinity. The former integral is is number which you ca easily obtaoin using contour integration methods. The latter you compute by doing the substitution x^2 = u as suggested by Hootenanny, and then you do a relpeated partial integration, where you integrate the sin and differentiate the 1/sqrt(u). You iterate this, each time integrating the trigonometric term and differentiating the 1/u^(n+1/2). This then yields an asymptotic expansion with the last unevaluated integral as an error term.

Last edited: Oct 28, 2009
15. Oct 28, 2009

### lurflurf

There is no systematic way to compute sine as I know.
But there are several approximation methods
thus it would be quite a surprize if fresnel integrals were easier