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Thread: 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.

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

    The information I have:

    1) $\displaystyle x=x(t)$

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

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

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

    (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.

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

    The information I have:

    1) $\displaystyle x=x(t)$

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

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

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

    (3 can be deduced from 2 or vice versa)

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

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

    Of course you won't be able to go any further without knowing what x is.
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    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, $\displaystyle df/dx=e^t$ doesn't make sense given that the RHS is a function of $\displaystyle t$. Also, condition (2) tells you nothing. The integral cannot be simplified as stated since you've given no info about the function $\displaystyle x(t)$.
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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, $\displaystyle df/dx=e^t$ doesn't make sense given that the RHS is a function of $\displaystyle t$. Also, condition (2) tells you nothing. The integral cannot be simplified as stated since you've given no info about the function $\displaystyle x(t)$.
    It makes sense because of the first condition x=x(t)

    To spell it out:

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

    (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. $\displaystyle f$ is a function of what variable? Where did the question come from? I suspect you've misinterpreted something.
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