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Math Help - definite integrals, upper limit variable

  1. #1
    Newbie Gustavis's Avatar
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    definite integrals, upper limit variable

    Hello

    it is correct to write:

    \int^x_0 f(x)\,dx

    or i need write ?

    \int^x_0 f(t)\,dt

    why ?

    The question is :
    if I need a substitution of f(x) in a formula :

    f(x)=sin(x)

    y=f(x) + \int^x_0 f(x)\,dx

    I can directly replace f(x)

    otherwise :

    y=f(x) + \int^x_0 f(t)\,dt

    don't seem me elegant imho..

    Thanks for the possible answers !
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by Gustavis View Post
    Hello

    it is correct to write:

    \int^x_0 f(x)\,dx

    or i need write ?

    \int^x_0 f(t)\,dt Mr F says: This is the one you want.

    why ? Mr F says: Let me re-write your two choices and then you tell me which one doesn't make sense:

    {\color{red}\int^{x = x}_{x = 0} f(x)\,dx}

    {\color{red}\int^{t = x}_{t = 0} f(t)\,dt}

    The question is :
    if I need a substitution of f(x) in a formula :

    f(x)=sin(x)

    y=f(x) + \int^x_0 f(x)\,dx

    I can directly replace f(x)

    otherwise :

    y=f(x) + \int^x_0 f(t)\,dt

    don't seem me elegant imho..

    Thanks for the possible answers !
    ..
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  3. #3
    MHF Contributor

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    Quote Originally Posted by Gustavis View Post
    Hello

    it is correct to write:

    \int^x_0 f(x)\,dx

    or i need write ?

    \int^x_0 f(t)\,dt
    As Mr. Fantastic indicatses, the second is far less confusing. In the first, you would have to recognize that you are using "x" to mean two different things- and that's never a good idea.

    why ?

    The question is :
    if I need a substitution of f(x) in a formula :

    f(x)=sin(x)

    y=f(x) + \int^x_0 f(x)\,dx

    I can directly replace f(x)

    otherwise :

    y=f(x) + \int^x_0 f(t)\,dt

    don't seem me elegant imho..

    Thanks for the possible answers !
    Would you say that,, when x= 1, y(1)= f(1)+ \int_0^1 f(1)d1? That doesn't even make sense: there is no such thing as "d1"! If you write instead y(1)= f(1)+\int_0^1 f(x)dx then you are recognizing that the "x" outside the integral and in the upper limit is NOT the same as the "x" inside the integral- and so you shouldn't use the same symbol!
    Far better to write y(x)= f(x)+ \int_0^x f(t)dt so that when x= 1, y(1)= f(1)+ \int_0^1 f(t)dt
    Last edited by mr fantastic; February 14th 2009 at 03:58 AM. Reason: Fixed quote tag
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