Question

solve the following integral

Answer 1

\[\Huge \int\limits_{0}^{\infty} \sin(x^2)dx\]


You can only Use Fourier or Laplce integrals to evaluate the above Integral . numerical integration is not required

Answer 2

@amistre64

Answer 3

im not uptodate with a fourier .... ive been meaning to teach myself that :) and im not sure that a laplace transform would be that applicable

Answer 4

fourier seems the what to go tho

Answer 5

2pi ift?

Answer 6

Could we use the Taylor's expansion of sin x^2 and then try to solve it? not sure if it'd converge

Answer 7

that would be a numerical approach i think

Answer 8

I tried that one but mentioned in text only laplace or fourier shall be used :(

Answer 9

oh there could be some Fourier transform property which might be able to use. I'll just check

Answer 10

\[\int_{0}^{\inf}sin(x^2)e^{-sx}~dt\]
derivatives of sin are easier to by parts than integrals

Answer 11

but then ... fourier just seems more applicable still

Answer 12

did you use the laplace for that @amistre64 ??

Answer 13

yes, that's LT

Answer 14

it'd be dx

Answer 15

yes, dx.

Answer 16

I'm thinking that you take the integral of the laplace of sine

Answer 17

I wonder how we can use this. http://www.wolframalpha.com/input/?i=laplace+transform+of+sin+%28t%5E2%29

Answer 18

if i rememer correctly, it's \(\frac{x^2}{s^2+(x^2)^2}\)

Answer 19

then, integrate using laplace definition.

Answer 20

that's for sin tx, we have x^2 inside.

Answer 21

I think you can use \(x^2\) since you're integrating with respect to "x"

Answer 22

?

Answer 23

or sine is \(\frac{1}{s^2+(x^2)^2}\) my mistake.

Answer 24

Here we have x instead of t
\[LT( \sin at)=\frac {a^2}{s^2+a^2}\]
\[LT( \sin t^2)\ne\frac {1^2}{s^2+1^2}\]

Answer 25

hm...idk. this math is far beyond me. i only went up to algebra.

Answer 26

@saadi What's the topic you are studying?

Answer 27

elementary school drop out.

Answer 28

I am studying fourier Integrals and applications from Advanced Engineering Mathematics by Erwin Kreszig .

Answer 29

Is there any similar looking question, worked out in the textbook?

Answer 30

no :(

Answer 31

MAPLE says Fresnel integral.

Answer 32

Plz guys don't give up on this I have exam day after tomorrow and really don't know how to deal with such questions :(

Answer 33

look up Fresnel integral.

Answer 34

the problem i posted is somewhat similar to yours and I solved it very similarly to how they did in the book.

Answer 35

@satellite73 @amistre64

Answer 36

refer this, should help. I'm also reading it
http://mcs.cankaya.edu.tr/proje/2010/guz/tahirokankale/rapor.pdf