Is there a way to simplify this integral involving an exponential function?

[BillKet]

Hello! I have a function ##f(t)## such that ##\int_a^b{f(t)dt}=f_0##. Is there a way to calculate (or bring it to a simpler form) ##\int_a^b{f(a)e^{t}}dt##? Thank you!

 

[trees and plants]

May i ask : does the exercise say that ##f(t)## is differentiable? the second integral contains##f(a)##?or perhaps it is ##f(t)##? what is your effort so far?

 

[etotheipi]

Hello! I have a function ##f(t)## such that ##\int_a^b{f(t)dt}=f_0##. Is there a way to calculate (or bring it to a simpler form) ##\int_a^b{f(a)e^{t}}dt##? Thank you!

As written, ##\int_a^b{f(a)e^{t}}dt = f(a)[e^b - e^a]##.

 

[BillKet]

As written, ##\int_a^b{f(a)e^{t}}dt = f(a)[e^b - e^a]##.

Ah sorry, the questions should be about ##\int_a^b{f(t)e^{t}}dt##

 

[BillKet]

May i ask : does the exercise say that ##f(t)## is differentiable? the second integral contains##f(a)##?or perhaps it is ##f(t)##? what is your effort so far?

It should be ##f(t)##. It is not an exercise, it is something obtained from a physics experiment, but I would say that yes, the function is differentiable. I can't say that I had many ideas, I was hoping there is probably some formula I don't know about that can be applied, as there is not much to do here with basic integration techniques.

 

[Mark44]

Ah sorry, the questions should be about ##\int_a^b{f(t)e^{t}}dt##

I don't think the integral can be evaluated without knowing more about f(t).

 

[trees and plants]

You could use the fundamental theorem of calculus for the first integral to get values for ##a## and ##b## for the antiderivative, let us say it ##F##, so you have ##F(b)-F(a)=f_0##.

We are talking here about definite integrals so for the second integral, someone i think would need to express it in terms of ##F(a),F(b)##, using integration by parts i think does not lead to computing it.

I want to say that there are ways to compute some integrals involving exponentials, polynomials, n-th roots, square roots and others, but here ##f(t)## is given without saying whether it has other properties or is of a form as for example the ones i said in this post.