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Math Help - Proving linearity property of the integral.

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    Proving linearity property of the integral.

    I have little problems to solve this it would be great if some body could help me.
    Last edited by chirkowatson; May 17th 2010 at 05:07 AM.
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    Quote Originally Posted by chirkowatson View Post
    I have little problems to solve this it would be great if some body could help me.
    What do you have to work with? Do you have any "rules" of integration or are you to prove this from the Riemann sums definition?
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    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by chirkowatson View Post
    I have little problems to solve this it would be great if some body could help me.
    There are two parts to this proof

    Part 1,

     \int [ f(x) + g(x) ] dx = \int f(x)dx  + \int g(x) dx


    Part 2,

     \int af(x) dx = \int a f(x) dx

    Let us compute part 1 first,

     \int [ af + bg](x) dx = \int [af(x) + bg(x)] dx

    The above doesn't need to be proved, it is simple expansion.

    Let  F(x) be the anti-derivative of  f(x) and let  G(x) be the anti-derivative of  g(x)

    In other words,

     F`(x) = f(x) and  G`(x) = g(x)

    Noting that  [F(x) + G(x)]` = F`(x) + G`(x)

    This is produced by both of the following,

     \int f(x)dx + \int g(x)dx =  [F(x) + G(x)]` + c = F`(x) + G`(x) + c

    and

     \int [ f(x) + g(x) ]dx = F`(x) + G`(x) + c

    Thus,

     \int [ f(x) + g(x) ] = \int f(x)dx + \int g(x)dx

    But I didn't include the constants, here. So what we are left with from the question is

     \int af(x)dx + \int bg(x)dx

    Let  F(x) be the anti-derivative of  f(x)

    Then,

     [aF(x)]` = aF`(x) = af(x)

    So,

     \int af(x) dx= aF`(x) + c = a \int f(x) dx
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