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Math Help - Integration

  1. #1
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    Integration

    Can someone please check that my answer for this question is correct:
    Question:
    <br />
A_n = \frac{2}{3} \int^{3}_{0} \frac{1}{2}x sin(\frac{n \Pi}{3} x) dx<br />

    My working out:
    <br />
A_n = \frac{1}{2} \int^{3}_{0} x sin(\frac{n \Pi}{3} x) dx
     = \frac{1}{2} {[x \frac{3}{n \Pi} cos (\frac{n \Pi}{3} x)]^{3}_{0} +\frac{3}{n \Pi} \int^{3}_{0} cos (\frac{n \Pi}{3} x) dx}
     = \frac{1}{2} {[\frac{-9}{n \Pi} cos (n \Pi) - 0] + \frac{3}{n \Pi} [\frac{3}{n \Pi} sin (\frac{n \Pi}{3} x)]^{3}_{0}}
     = \frac{9}{2 n^2 \Pi^2}sin(n \Pi) - \frac{9}{2 n \Pi} cos (n \Pi)
     = \frac{9}{2 n \Pi} (-1)^(n+1) - frac{9}{2 n \Pi} cos (n \Pi)

    for some reason i think this looks wrong, can someone please confirm with me? thank you
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by gconfused View Post
    Can someone please check that my answer for this question is correct:
    Question:
    <br />
A_n = \frac{2}{3} \int^{3}_{0} \frac{1}{2}x sin(\frac{n \Pi}{3} x) dx<br />

    My working out:
    <br />
A_n = \frac{1}{2} \int^{3}_{0} x sin(\frac{n \Pi}{3} x) dx
     = \frac{1}{2} {[x \frac{3}{n \Pi} cos (\frac{n \Pi}{3} x)]^{3}_{0} +\frac{3}{n \Pi} \int^{3}_{0} cos (\frac{n \Pi}{3} x) dx}
     = \frac{1}{2} {[\frac{-9}{n \Pi} cos (n \Pi) - 0] + \frac{3}{n \Pi} [\frac{3}{n \Pi} sin (\frac{n \Pi}{3} x)]^{3}_{0}}
     = \frac{9}{2 n^2 \Pi^2}sin(n \Pi) - \frac{9}{2 n \Pi} cos (n \Pi)
     = \frac{9}{2 n \Pi} (-1)^(n+1) - frac{9}{2 n \Pi} cos (n \Pi)

    for some reason i think this looks wrong, can someone please confirm with me? thank you
    second line multiply with 1/3 not 1/2
    what you do in the last step
    other things is wright
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  3. #3
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    Quote Originally Posted by Amer View Post
    second line multiply with 1/3 not 1/2
    what you do in the last step
    other things is wright
    thank you for helping

    well this is actually a differential equation question, but i just needed help with the integration part.. Umm i dont really know what i did in the last step

    coz that's the way they simplified it in my lecture notes for partial differential equations, but yeahh i probably did that wrong too ><
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  4. #4
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    ohh sorries the last line was:


    = \frac{9}{2 n \Pi} (-1)^{n+1} - \frac{9}{2 n \Pi} cos (n \Pi)<br />

    but i don't know if that's how you simplify it
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  5. #5
    MHF Contributor Amer's Avatar
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    Ok


    suppose it was correct lats simplify it sin(n pi ) = 0 for all n natural number


    \frac{9}{2n\pi}cos(n\pi)=\begin{cases} \left(\frac{9}{2n\pi}\right)    \mbox{ if } n..even \\ -\left(\frac{9}{2n\pi}\right)       \mbox{ if } n..odd \\\end{cases}
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