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    I'm really struggling with an integration that i'm sure is quite simple but i seem to be going round in circles


    f(x)=\displaystyle \int^1_0 tx(e^t-\dfrac{x}{6})dt


    Taking u=e^x -\dfrac{x}{6}.
    Then u'=e^x-\dfrac{1}{6}


    Then v'=tx,Hence v= \dfrac{xt^2}{2}


    By integration by parts: uv- \int vdu


    (e^x-\dfrac{x}{6})(\dfrac{xt^2}{2})\ -\ \displaystyle \int^1_0 \dfrac{xt^2}{2}(e^x-\dfrac{1}{6})


    Which requires integration by parts etc... where am i going wrong?
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    (Original post by gemma331)
    where am i going wrong?
    From the beginning.

    Start by expanding the brackets.

    One term is a straight integration, and the other requires IBP.

    Switch your u and v around so v' = e^t.
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    (Original post by gemma331)
    I'm really struggling with an integration that i'm sure is quite simple but i seem to be going round in circles


    f(x)=\displaystyle \int^1_0 tx(e^t-\dfrac{x}{6})dt


    Taking u=e^x -\dfrac{x}{6}.
    Then u'=e^x-\dfrac{1}{6}


    Then v'=tx,Hence v= \dfrac{xt^2}{2}


    By integration by parts: uv- \int vdu


    (e^x-\dfrac{x}{6})(\dfrac{xt^2}{2})\ -\ \displaystyle \int^1_0 \dfrac{xt^2}{2}(e^x-\dfrac{1}{6})


    Which requires integration by parts etc... where am i going wrong?
    I'm with Ghostwalker here.

    By the way, it's not 100% clear which of x and t is the constant term, but I'm assuming it's x.

    If you can evaluate

    \displaystyle \int xe^x \ dx

    then there's not that much more to be done here.

    If you can't

    Spoiler:
    Show


    take

    u = x

    and

    v' = e^x

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    (Original post by gemma331)
    I'm really struggling with an integration that i'm sure is quite simple but i seem to be going round in circles


    f(x)=\displaystyle \int^1_0 tx(e^t-\dfrac{x}{6})dt


    Taking u=e^x -\dfrac{x}{6}.
    Then u'=e^x-\dfrac{1}{6}


    Then v'=tx,Hence v= \dfrac{xt^2}{2}


    By integration by parts: uv- \int vdu


    (e^x-\dfrac{x}{6})(\dfrac{xt^2}{2})\ -\ \displaystyle \int^1_0 \dfrac{xt^2}{2}(e^x-\dfrac{1}{6})


    Which requires integration by parts etc... where am i going wrong?
    If your integration variable is t, then you can treat x as a constant as far as the integration is concerned, so multiply out your bracket into 2 terms and take any 'x factors' outside the integral. The only thing you have left that requires integration by parts is te^t
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    You're integrating with respect to t, not x. In the second integral you can treat all the non-t factors as constants which can be taken outside, so all you then have to do is integrate t^2 dt.
 
 
 
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