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Anti derivative

  1. Aug 22, 2008 #1
    whats the anti derivative of cos(theta^2)
    do i use the chain rule
  2. jcsd
  3. Aug 22, 2008 #2


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    somebody could answer this and make me look stupid but im going to guess this is a function whose antiderivative cannot be expressed as an elementary combination of the usual suspects.

    if you think about it, using the chain rule would be expected to give you a product as an answer, so it is hard to imagine how to get this function as a derivative.

    of course it is continuous, hence the ftc says it is the derivative of its indefinite integral.
  4. Aug 22, 2008 #3
    so i find the derivative of it
  5. Aug 22, 2008 #4
  6. Aug 22, 2008 #5
    i thought it would be 2* sin (theta)^2
  7. Aug 22, 2008 #6


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    the anti-derivative of [itex]cos(\theta^2)[/itex] can't be expressed in terms of elementary functions as mathwonk said.

    But if you want to find the derivative of it, you'll need the chain rule.
  8. Aug 22, 2008 #7
    how would i find the antiderivative of it
  9. Aug 23, 2008 #8


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    Not all functions have anti-derivatives that can expressed in familiar form (ie. elementary form) This would be one of those.
  10. Aug 23, 2008 #9
    Look up the MacLaurin of [tex]cos(\theta)[/tex], from there figure out what the MacLaurin would be for [tex]cos(\theta^{2}) [/tex]. Now you can integrate it out and see what you get.
  11. Aug 23, 2008 #10
    Whenever a function is difficult or even impossible to integrate in terms of the elementary functions, you can always reduce it to an infinite series and integrate it term by term.

    The series for cos (x) is 1 – x^2/2! + x^4/4! – x^6/6! . . . . .

    Just plug x^2 in there in place of x and get:

    1 – x^4/2! + x^8/4! – x^12/6! . . . . .

    Now just integrate that term by term. Because of the factorials in the denominator the series converges quickly so only 3 terms should be needed unless you require great accuracy.
  12. Aug 23, 2008 #11
    ok i get it, i'm very slow like a turtle,

    "The mind may be slow at times, but through time the information will be gathered."
    Eugeno Ponce
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