Solution to this integral?

  • Thread starter Machinus
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  • #1
Machinus
[tex]\int cos(x^2)dx[/tex]
 

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  • #2
dextercioby
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Not an elementary function.Search the same site (A&S online,see the other thread) for FRESNEL INTEGRALS.

Daniel.
 
  • #3
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forget it, this one has no elementary solution.. you can espand the cosine by Taylor series... and integrate the individual term...
 
  • #4
dextercioby
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However,the series obtained has a very small convergence radius.It would virtually do you no good.If u have definite integrals involving C(x),then learn they are tabulated.

Daniel.
 
  • #5
Galileo
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The power series expansion is the basic surefire way to get a numerical approximation.

[tex]\cos (x) = \sum_n^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}[/tex]

[tex]\cos (x^2) = \sum_n^{\infty}\frac{(-1)^nx^{4n}}{(2n)!}[/tex]

[tex]\int \cos (x^2)dx = \sum_n^{\infty}\frac{(-1)^nx^{4n+1}}{(4n+1)(2n)!}+C[/tex]

Since the power series for [itex]\cos(x)[/itex] converges for all x, so do the power series for [itex]\cos(x^2)[/itex] and [itex]\int \cos(x^2)dx[/itex].
It may take some computational power if you're interested in values of x that are far from 0.
 
  • #6
dextercioby
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Please,Galileo,compute using your formula
[tex] C(8)=...? [/tex]
,defining
[tex] C(x)=:\int_{0}^{x} \cos(t^{2}) dt [/tex]

Daniel.
 
  • #7
Galileo
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Approximately 0.68396
 
  • #8
dextercioby
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Interesting...How many terms did u add??You couldn't have added them all...


Daniel.
 
  • #9
Galileo
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I just used maple to sum the thing from n=0 to n=100.
That number rounded to 10 decimal places is: [itex]C(8) \approx 0.6839570275[/itex].

That the series converges for all x follows from the ratio test for example.
 

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