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Math Help - Is there anything can not be integrated or differentiated?

  1. #1
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    Is there anything can not be integrated or differentiated?

    My teacher was saying that you can not integrate e^x^2(with no limits).He did say something can not be integrated but then later in the course it turned out you can do it by parts.So Is this true?or there is a way that he is not telling me?
    Thank you.
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  2. #2
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    e^{x^2} is a continuous function. Therefore it is integrable. It's integral cannot however be expressed in terms of elementary functions.

    There are lots of functions that cannot be integrated at all.
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  3. #3
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    Can I see some functions that can not be integrated? He likes to ask me questions and if i dont know the answer,he likes to stare into my soul with this look...... :S
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  4. #4
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    I assume we're talking about Riemann integration here. Any function that can't be integrated must be messy because all piecewise continuous functions are integrable.

    Here's a classic example: Let f(x) = 0 if x is rational, and f(x) =1 if x is irrational. Then the lower sums converge to 0 and the upper sums converge to 1, so that the integral does not exist.
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  5. #5
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    There are functions f(x) such that there is a function F(x) such that F'(x)=f(x) but yet f(x) is not integrable.

    In other words, there are derivatives that have no integral.
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