# Simplifying an integral

• I
• BillKet
In summary, the conversation discusses the possibility of calculating or simplifying the integral ##\int_a^b{f(t)e^{t}}dt##, where ##f(t)## is a given function and ##a## and ##b## are constants. It is mentioned that the function ##f(t)## is differentiable and obtained from a physics experiment, and that basic integration techniques may not be applicable. It is suggested to use the fundamental theorem of calculus for the first integral, but it is unclear how to proceed with the second integral without more information about ##f(t)##. It is also mentioned that there are methods for computing certain types of integrals, but it is uncertain if they would apply to this particular integral.f

#### 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!

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?

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]##.

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##

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.

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).

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.