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Math Help - Differentiate an integral

  1. #1
    Oli
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    Differentiate an integral

    Anybody know how to differentiate:

    Int[0,x](f(k)(x-k))dk

    with respect to x?
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  2. #2
    Oli
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    No worries:
    Differentiation under the integral sign - Wikipedia, the free encyclopedia

    I may be the first person to get a degree using nothing but wikipedia...
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  3. #3
    Super Member angel.white's Avatar
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    Quote Originally Posted by Oli View Post
    No worries:
    Differentiation under the integral sign - Wikipedia, the free encyclopedia

    I may be the first person to get a degree using nothing but wikipedia...
    Wikipedia is usually over my head I usually use "brute force" where I try every possible method until I eventually arrive at a correct answer, then I try that method on the next question and see if it works :P

    On the down side, it probably took me 80 hours to figure out how to differentiate. >.<
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  4. #4
    Moo
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    Hello,

    Actually, this is not so difficult oO

    If F is a primitive of f, we have :

    \int_a^b f(x) dx = F(b)-F(a)

    So transform your integral with it. Then, differentiate it...
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Unless

    Quote Originally Posted by Oli View Post
    Anybody know how to differentiate:

    Int[0,x](f(k)(x-k))dk

    with respect to x?
    I am misunderstanding the question I think what you want is the second fundamental theorem of calculus with states \frac{d}{dx}\bigg[\int_a^{x}f(t)dt\bigg]=f(x)
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  6. #6
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    Quote Originally Posted by Mathstud28 View Post
    I am misunderstanding the question I think what you want is the second fundamental theorem of calculus with states \frac{d}{dx}\bigg[\int_a^{x}f(t)dt\bigg]=f(x)
    No - because the integrand is also a function of x, the variable of differentiation.

    The question requires \frac{d}{dx}\bigg[\int_a^{x}f(t, x) \, dt \bigg].
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  7. #7
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    Quote Originally Posted by Moo View Post
    Hello,

    Actually, this is not so difficult oO

    If F is a primitive of f, we have :

    \int_a^b f(x) dx = F(b)-F(a)

    So transform your integral with it. Then, differentiate it...
    No. See post #6.
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  8. #8
    MHF Contributor arbolis's Avatar
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    In your case the borns of the integral are 0 and x. Thefore, for a consequence of some theorem (that I don't remember exactly), the derivative of your integral is equal to what is under the integral. Notice that your integral is dt, not dx...
    Imagine you had the same integral, but with the upper born equal to x^2, then differentiate your integral, respect to x would be f(k)(x^2-k)(2x), it's a consequence of the chain rule I think.
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  9. #9
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    Quote Originally Posted by arbolis View Post
    In your case the borns of the integral are 0 and x. Thefore, for a consequence of some theorem (that I don't remember exactly), the derivative of your integral is equal to what is under the integral. Notice that your integral is dt, not dx...
    Imagine you had the same integral, but with the upper born equal to x^2, then differentiate your integral, respect to x would be f(k)(x^2-k)(2x), it's a consequence of the chain rule I think.
    Sorry, but you're wrong. The reason is given in post #6.
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  10. #10
    MHF Contributor arbolis's Avatar
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    Ok I'm probably wrong, but reading my class notes on calculus, I found :
    If h is continue, f and g derivable and F(x)= Integral from f(x) to g(x) of h(t) dt, then F'(x)=h(g(x))g'(x)-h(f(x))f'(x).
    With this formula, you can calcul the derivative of your problem. And this formula is true.
    Hope this helps, at least for further derivatives of integrals involving 2 borns in function of x.
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  11. #11
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    Quote Originally Posted by arbolis View Post
    Ok I'm probably wrong, but reading my class notes on calculus, I found :
    If h is continue, f and g derivable and F(x)= Integral from f(x) to g(x) of h(t) dt, then F'(x)=h(g(x))g'(x)-h(f(x))f'(x).
    With this formula, you can calcul the derivative of your problem. And this formula is true.
    Hope this helps, at least for further derivatives of integrals involving 2 borns in function of x.
    Your formula is correct but is NOT valid for this problem. As has been said in post #6, the integrand is NOT h(t). It's h(t, x), that is, it's a function of t (the dummy variable of integration) AND x (the variable of differentiation).
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  12. #12
    MHF Contributor arbolis's Avatar
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    I'm sorry, I missed it.
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  13. #13
    Moo
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    Hm, so it has to deal with convergence or something like that ?
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  14. #14
    Oli
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    Guys! Thanks for help but its Okay!

    Look at the wiki article I put a link to in post 2.

    It has pretty much this exact function.
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  15. #15
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    Quote Originally Posted by Oli View Post
    Guys! Thanks for help but its Okay!

    Look at the wiki article I put a link to in post 2.

    It has pretty much this exact function.
    I know that and you know that. But if incorrect posts are made, they have to be addressed. Otherwise others who read this thread might be mislead. There is clearly some confusion on:
    1. The significnace of the integrand containing the variable of differentiation.
    2. When the FVC can be applied.

    This confusion has to be pointed out and addressed.
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