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Math Help - Is it possible to integrate dx/dt*e^t in a nice and compact way?

  1. #1
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    Is it possible to integrate dx/dt*e^t in a nice and compact way?

    Hello,

    I need to integrate this based on certain information. I've tried integration by parts, looked in tables, etc., and am pretty sure there's just no neat way to do it. But I want to check with the forum before I give up.

    \int \frac{dx}{dt} e^t \: dt

    The information I have:

    1) x=x(t)

    2) \frac{d\ln{f_x}}{dt}=1

    (where the notation f_x=\frac{df}{dx})

    3) \frac{df}{dx}=e^t

    (3 can be deduced from 2 or vice versa)

    Thanks
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  2. #2
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    Re: Is it possible to integrate dx/dt*e^t in a nice and compact way?

    Quote Originally Posted by rainer View Post
    Hello,

    I need to integrate this based on certain information. I've tried integration by parts, looked in tables, etc., and am pretty sure there's just no neat way to do it. But I want to check with the forum before I give up.

    \int \frac{dx}{dt} e^t \: dt

    The information I have:

    1) x=x(t)

    2) \frac{d\ln{f_x}}{dt}=1

    (where the notation f_x=\frac{df}{dx})

    3) \frac{df}{dx}=e^t

    (3 can be deduced from 2 or vice versa)

    Thanks
    Are you sure you can't use integration by parts? Let \displaystyle \begin{align*} u = e^t \implies du = e^t\,dt \end{align*} and \displaystyle \begin{align*} dv = \frac{dx}{dt}\,dt \implies v = \int{\frac{dx}{dt}\,dt} = \int{dx} = x \end{align*}, giving

    \displaystyle \begin{align*} \int{\frac{dx}{dt}\,e^t\,dt} = x\,e^t - \int{x\,e^t\,dt} \end{align*}

    Of course you won't be able to go any further without knowing what x is.
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  3. #3
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    Re: Is it possible to integrate dx/dt*e^t in a nice and compact way?

    Yes that's what I got when using integration by parts, but I'd like to get something that doesn't have another integral in it.
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    Re: Is it possible to integrate dx/dt*e^t in a nice and compact way?

    There's something wrong with the info you've provided. Firstly, df/dx=e^t doesn't make sense given that the RHS is a function of t. Also, condition (2) tells you nothing. The integral cannot be simplified as stated since you've given no info about the function x(t).
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    Re: Is it possible to integrate dx/dt*e^t in a nice and compact way?

    Quote Originally Posted by ojones View Post
    There's something wrong with the info you've provided. Firstly, df/dx=e^t doesn't make sense given that the RHS is a function of t. Also, condition (2) tells you nothing. The integral cannot be simplified as stated since you've given no info about the function x(t).
    It makes sense because of the first condition x=x(t)

    To spell it out:

    \frac{d\ln{f_x}}{dt}=1 \Rightarrow \int d\ln{f_x}=\int dt \Rightarrow \ln{f_x}=t+C  \Rightarrow f_x=e^{t+C}

    (where t=t(x))
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  6. #6
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    Re: Is it possible to integrate dx/dt*e^t in a nice and compact way?

    The question is not posed properly. f is a function of what variable? Where did the question come from? I suspect you've misinterpreted something.
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