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Math Help - tricky (?) intergration with inclusion of arcsin(x-3)

  1. #1
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    tricky (?) intergration with inclusion of arcsin(x-3)

    i am getting myself a bit lost with the integration of a function P(x), i have tried to solve it by integration by parts, but then end up needing to integrate arcsin(x-3), which is proving trickier than i was expecting, i have come up with an answer but its not give the results that i expect, i'e between the limits of 2 and 4 the answer should be 3. would anyone mind looking at the attached PDF and give me a few pointers where i am going wrong
    Cheers
    WN
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  2. #2
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    Hi

    Let u = 3-x

    \int \frac{x}{\sqrt{1-(3-x)^2}}\:dx = \int \frac{u-3}{\sqrt{1-u^2}}\:du

    \int \frac{x}{\sqrt{1-(3-x)^2}}\:dx = \int \frac{u}{\sqrt{1-u^2}}\:du - 3 \int \frac{1}{\sqrt{1-u^2}}\:du = -\sqrt{1-u^2} - 3 Arcsin u
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  3. #3
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    Quote Originally Posted by running-gag View Post
    Hi

    Let u = 3-x

    \int \frac{x}{\sqrt{1-(3-x)^2}}\:dx = \int \frac{u-3}{\sqrt{1-u^2}}\:du

    \int \frac{x}{\sqrt{1-(3-x)^2}}\:dx = \int \frac{u}{\sqrt{1-u^2}}\:du - 3 \int \frac{1}{\sqrt{1-u^2}}\:du = -\sqrt{1-u^2} - 3 Arcsin u
    sorry you have lost me, where does this come in to it, is this the whole answer, the answer after the second intergration by parts, or where i am doing the substitution at the end. i can't seem to find where is it.
    also you have u=3-x, then in the second equation you have x=u-3 which isn't right as from the definition on u, x-3-u
    Hope you can clarify that
    Cheers
    WN
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  4. #4
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    I only used substitution (no integration by parts)

    u = 3-x gives x=3-u and dx=-du therefore x dx = (u-3) du

    \int \frac{x}{\sqrt{1-(3-x)^2}}\:dx = \int \frac{u-3}{\sqrt{1-u^2}}\:du

    Then I split the integral in 2 parts

    \int \frac{x}{\sqrt{1-(3-x)^2}}\:dx = \int \frac{u}{\sqrt{1-u^2}}\:du - 3 \int \frac{1}{\sqrt{1-u^2}}\:du = -\sqrt{1-u^2} - 3 Arcsin u
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  5. #5
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    Quote Originally Posted by running-gag View Post
    I only used substitution (no integration by parts)
    ah yes i see now, thanks a lot.
    WN
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