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Math Help - Integration question

  1. #1
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    Integration question

    Hi

    let g(x) =

    where f is an increasing function. Then,

    a. g{x} is an increasing function of x.
    b. g{x} is an decreasing function of x.
    c. g{x} is an increasing function of x>0 and decreasing for -10<x<10.
    d. none of the above.

    thanks
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  2. #2
    MHF Contributor Calculus26's Avatar
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    g ' (x) = x f '(x) (from 2d fundamental theorem)

    since f(x) is increasing f ' (x) > 0

    Therefore the answer is increasing if x > 0 and decreasing if x < 0

    Are you sure c is Right it doesn't make sense seems it should read:

    increasing x>0 and decrasing -10< x < 0
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  3. #3
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    hi!
    it depands on what kind of integration that you consider,if you consider the lebesgue integration, then you can choose f as a Cantor function,then f is increasing function,while f'=0almost everywhere.so g=0forever!
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  4. #4
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    thanks Can the Second Fundamental theorem be applied to Definate integrals also? I was under the impression that it was only for indefinite ones.
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  5. #5
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    and if you conside Riemann integration,just like what Calculus26 said!
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  6. #6
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    Quote Originally Posted by berlioz View Post
    hi!
    it depands on what kind of integration that you consider,if you consider the lebesgue integration, then you can choose f as a Cantor function,then f is increasing function,while f'=0almost everywhere.so g=0forever!
    i dont think we have done other types of integrals.
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  7. #7
    MHF Contributor Calculus26's Avatar
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    2d fundamental thm

    See attachment for statement of 2d FTC
    Attached Files Attached Files
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