Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solution to this integral?

  1. Jan 13, 2005 #1
    [tex]\int cos(x^2)dx[/tex]
     
  2. jcsd
  3. Jan 13, 2005 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

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

    Daniel.
     
  4. Jan 13, 2005 #3
    forget it, this one has no elementary solution.. you can espand the cosine by Taylor series... and integrate the individual term...
     
  5. Jan 13, 2005 #4

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  6. Jan 14, 2005 #5

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  7. Jan 14, 2005 #6

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Please,Galileo,compute using your formula
    [tex] C(8)=...? [/tex]
    ,defining
    [tex] C(x)=:\int_{0}^{x} \cos(t^{2}) dt [/tex]

    Daniel.
     
  8. Jan 14, 2005 #7

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    Approximately 0.68396
     
  9. Jan 14, 2005 #8

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Interesting...How many terms did u add??You couldn't have added them all...


    Daniel.
     
  10. Jan 14, 2005 #9

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Solution to this integral?
Loading...