Hi...I'm trying to solve a problem involving integrating factors.
Whilst using the integrating factor method, I am required to integrate a function multipled by another function.
say f(t) = exp(kt) and some other function g(t); where exp = exponential and k is some constant.
∫ f(t)*g(t) dt or
∫ exp(kt)*g(t) dt
What would the result of this integral be? I have never met an integral like this before. Would it simply be exp(kt)*g(t)/k?
More specifically, the problem and my attempted answer is in PDF format:
In my attempted solution, I am unsure about the last two lines I have written out, as it relates to integrating a function multipled by another function.
Many thanks in advance.
Hi,
In your solution there is an error in the third line. (I am going to use Int[f(x) dx] for the integral of f(x).] You have
p(t) = exp(Int[ R/L t] dt) = exp( R/L t)
You should have
p(t) = exp(Int[ R/L t] dt) = L/R exp( R/L t)
so in the remainder of the calculation you are missing a factor of L/R.
When you get to the line which is third from the end the factors Vo and L are constants so you can bring them outside the integral and you then have to find
Int[exp( R/L t) (1 - exp( -t/T)) dt]
= Int[exp( R/L t) - exp( R/L t) exp(- 1/T t) dt]
= Int[exp( R/L t) - exp( R/L t - 1/T t) dt]
= Int[exp( R/L t) - exp(( R/L - 1/T )t) dt]
= Int[exp( R/L t) - exp(( RT - L/LT t) dt]
= Int[exp( R/L t) dt] - Int[exp(( RT - L/LT t) dt]
= L/R exp( R/L t) - LT/RT - L exp(( RT - L/LT t) + C
Harley
Dear Harley,
In your response to my question you said that my third line of working out is wrong. It seems that you have misread the line as you said I had
p(t) = exp(Int[ R/L t] dt) = exp( R/L t)
where in fact I had
p(t) = exp(Int[ R/L] dt) = exp( R/L t)
If you read the third line of the PDF file again, you will see that the integral expression did not initally contain a t variable.
Thanks for the tip on taking the constants Vo and L outside of the integral expression..this simplifies the expression nicely. However, what totally slipped my mind was integration by parts....but taking your advice about taking the constants outside of the integrand, the whole integral seems to cancel down nicely...these are the steps I have taken
(Vo/L)*exp(-Rt/L) * [Int[ exp(Rt/L) * (1-exp(-t/T)) dt] + C]
let z = (Vo/L)*exp(-Rt/L); for now.
z * [Int[ exp(Rt/L) * (1-exp(-t/T)) ] dt + C]
Expand out the brackets
z * [Int[ exp(Rt/L) - exp(Rt/L - t/T) ] dt+ C]
Seperate exp(Rt/L - t/T) term into two separate exponentials
z * [Int[ exp(Rt/L) - exp(Rt/L) + exp(t/T) ] dt + C]
Collect/cancel terms
z * [Int[ exp(t/T) ] dt + C]
So....resubstituting back for z....
(Vo/L)*exp(-Rt/L)*[Int[ exp(t/T) ] dt + C]
I am correct in doing that after taking your advice about taking the constants Vo and L out of the integral expression?
Thanks again....David
Hi David,
You are correct that I misread your third line.
There is however an error in your reply. You have
Expand out the brackets
z * [Int[ exp(Rt/L) - exp(Rt/L - t/T) ] dt+ C]
Seperate exp(Rt/L - t/T) term into two separate exponentials
z * [Int[ exp(Rt/L) - exp(Rt/L) + exp(t/T) ] dt + C]
In the second term you have
- exp(Rt/L - t/T)
which you then rewrote
- exp(Rt/L) + exp(t/T)
You need to remember that these are actually exponents, in the more usual notation
- exp(Rt/L - t/T) = - eRt/L - t/T
which is not the same as
- exp(Rt/L) + exp(t/T) = - eRt/L + et/T
What you can do with the expression
- exp(Rt/L - t/T) = - eRt/L - t/T
is to notice that t is a common factor in the exponent so
- exp(Rt/L - t/T) = - eRt/L - t/T = - e(R/L - 1/T)t
and R/L - 1/T is a constant. Hence your expression
z * [Int[ exp(Rt/L) - exp(Rt/L - t/T) ] dt+ C]
can be rewritten
z * Int[eA t - eB t ] dt + C
where A and B are constants A = R/L and B = R/L - 1/T. Thus you get
z * [1/A eA t - 1/B eB t ] + C
Harley