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Math Help - Proving Question - Stucked

  1. #1
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    Proving Question - Stucked

    Proving Question - Stucked-question1.gif


    Hi, i am stucked with this question after stuck at double integral part. Any help would be appreciated. Thank you very much
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  2. #2
    Ted
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    Re: Proving Question - Stucked

    What about using the hint?
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  3. #3
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    Re: Proving Question - Stucked

    i used the hint but it doesnt seem to get me anywhere, i show u which part i stucked at lol
    after i scanned.......
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  4. #4
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    Re: Proving Question - Stucked

    wonder where the double integral came in here???
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  5. #5
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    Re: Proving Question - Stucked

    the double integral come when i integrate by parts:

    check this out for me:

    u = pi - x

    integrate from 0 to pi: xf(sin x) dx
    = integrate from pi to 0 (pi - u)f(sin (pi - u) ) du (sub u = pi -x)
    =integrate from pi to 0 uf(sin (pi - u) du - (pi)integrate from pi to 0 f(sin(pi - u) du
    = (u)integrate from pi to u f(sin (pi - u)) du - integrate from pi to 0 integrate from pi to 0 f(sin (pi - u)) du - (pi) integrate from pi to 0 f(sin(pi - u) du.
    the double integral is at the middle portion.
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  6. #6
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    Re: Proving Question - Stucked

    I see no reason to use integration by parts or any double integral.

    Let u= \pi- x as the hint suggests. Then du= -dx. Further, when x= 0, u= \pi and when x= \pi u= 0 so the limits of integration are swapped. But that "-" just swaps them back again. Of couse, [itex]x= \pi- u[/ietx]. In terms of the variable u, the integral is
    \int_0^\pi (\pi- u)f(sin(\pi- u) du= \pi\int_0^\pi f(sin(\pi- u))du- \int_0^\pi u f(sin(\pi- u))du

    And, of course, sin(\pi- u)= sin(u) so we have
    \int_0^\pi xf(sin(x))dx= \pi\int_0^\pi f(sin(u))du- \int_0^\pi uf(sin(u))du

    But these are definite integrals- the variables are "dummy" variables- they don't exist outside the integrals. So it is perfectly valid to make the new substitution "u= x" on the right and have.
    \int_0^\pi xf(sin(x))dx= \pi\int_0^\pi f(sin(x))dx- \int_0^\pi xf(sin(x))dx
    and the result follows easily.
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  7. #7
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    Re: Proving Question - Stucked

    Thanks alot haha, I didnt see that part
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