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

1. ## 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

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

Originally Posted by rainer
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.

3. ## 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.

4. ## 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 $\displaystyle x(t)$.

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

Originally Posted by ojones
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))

6. ## 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.