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Math Help - Derivative of an integral

  1. #1
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    Derivative of an integral

    Hi everyone,
    First of all, great forum! Hope someone out there can help me : )

    Here`s the problem:

    Let f(x) = sin( \frac{1}{x})  for x \neq 0, f(0) = 1

    Calculate the derivative of F(x) = \int_0^x f(t)dt
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  2. #2
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    You have to use the Calculus Fundamental Theorem: \left( \int_0^x f(t)dt \right) '=f(x)
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  3. #3
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    You mean the answer is just: sin( \frac{1}{x} )?
    (That`s way too easy)

    Anyway, thanks a lot for the quick reply : )

    Ok, one more question, so I don`t open a new thread:
    How can I proof that f(x) is integrable over every
    interval [a,b]?

    EDIT:
    I guess this helps a lot xD
    "If f(x) is bounded on [a,b] and has finitely many points of discontinuity on [a,b],
    then f is integrable on [a,b]."
    Last edited by rincewind2; February 5th 2010 at 11:51 PM.
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  4. #4
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    Quote Originally Posted by rincewind2 View Post
    You mean the answer is just: sin( \frac{1}{x} )?
    I think there's more to it - note that you are doing a definite integral from 0 to x. Do you know how to plug in your limits of integration into your integral? Remember that they told you what f(0) is, since sin(1/0) is undefined.

    Brian
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  5. #5
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    Quote Originally Posted by buckeye1973 View Post
    Do you know how to plug in your limits of integration into your integral?
    Brian
    I`m sorry, but no : (
    Can you explain more... I really need this one
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  6. #6
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    Quote Originally Posted by rincewind2 View Post
    I`m sorry, but no : (
    Can you explain more... I really need this one
    You are going to need to know how to calculate a definite integral to solve this problem, and if you are doing calculus, you really ought to know how to calculate a definite integral! This problem doesn't make sense without it.

    EDIT: Sorry, as I wasn't thinking right when I gave this answer, Vince's answer below is better.
    Last edited by buckeye1973; February 7th 2010 at 07:32 AM. Reason: oops, not correct!
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  7. #7
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    so in your case the answer is what felper said. I'm very sure.

    On-Edit: Everything on the right hand side in the equaiton above is 0 except the second term which is your answer. there is no need to worry about the limits because you're differentiating with respect to x not t. Had the integrand been a function of both x and t, then you would have had to worry about the limits.
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  8. #8
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    Lol... finally got it xD
    Everyone, thank you for the replies, you`ve been most helpfull : )
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