Integration of two exponential functions

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Michael Henry
Michael Henry 2017-4-3
评论： Star Strider 2017-4-4
Hello my friends, I have a problem with solving the following integral. It is a combination of two exponentials with different ratio in the attached picture.
I will be very grateful if you can help me with this. The derivation steps, if provided, will be much better to me as I have to apply this kind of integration in other problems I have.
If there is no closed form, can you please help me on how to integrate it using MATLAB with constants a, b, \lambda, and y to appear in the last answer.
Thanks in advance.. :)
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David Goodmanson
David Goodmanson 2017-4-3
Hi sharief, This doesn't look like a very tractable integral, but at least it's not oscillatory. If you make the substitution x -> x/b, dx -> dx /b then you end up with
C = (1/b) Integral{0,inf} exp((ay/b)/(1+x)) exp(-(lambda/b)x) dx
so this is really just a two-parameter integral, a function of ay/b and lambda/b. That's not so bad, so one approach would be to make a 2-d table by numerical integration and interpolate off of it, not forgetting to multiply by 1/b afterwards.
Michael Henry
Michael Henry 2017-4-3
Hello David.
I am really thankful for your idea. However, I am not interested in the value of integral itself. Actually, the next thing I am gonna do after this integration is to optimize a and b. Hence, what I am really interested at is to get a closed form for this integration. Can you please tell me if there is any way that I can use MATLAB to get a closed form expression for my integration above?

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Star Strider
Star Strider 2017-4-3
There does not appear to be an analytical solution. Constraining ‘a+b=1’ simply requires defining ‘a=1-b’, since ‘a’ only appears once.
This seems the best you can do:
syms b lambda x y
f(x) = exp(((1-b)/b)*y/(b*x+1)) * exp(-lambda*x);
F = int(f, x, 0, Inf);
F_fcn = matlabFunction(simplify(F,'Steps',10))
F_fcn =
function_handle with value:
@(b,lambda,y)integral(@(x)exp(-lambda.*x).*exp(-(y.*(b-1.0))./(b.*(b.*x+1.0))),0.0,Inf)
or more directly:
F_fcn = @(b,lambda,y) integral(@(x)exp(-lambda.*x).*exp(-(y.*(b-1.0))./(b.*(b.*x+1.0))),0.0,Inf);
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Michael Henry
Michael Henry 2017-4-4
Thank you so much for your help. I tried to run the code but I am getting the following errors
"
Error using mupadmex Error in MuPAD command: Can integrate only with respect to identifiers. [int]
Error in symfun/feval>evalScalarFun (line 69) y = mupadmex('symobj::zipargs',Ffun,inds{:});
Error in symfun/feval (line 33) y = evalScalarFun(F.fun, varargin);
Error in fplot (line 101) x = xmin; y = feval(fun,x,args{4:end});
Error in work (line 9) fplot(Fsym, [0 1E+2])
"
Can you please tell me what's wrong? I am sorry for my repeated questions but I am new with MATLAB and I need your experience please. Thank you..
Star Strider
Star Strider 2017-4-4
My pleasure.
I am not certain. The code I posted ran without error. The values of the parameters could be a problem, however I do not know the parameters you provided or the code you ran.

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Michael Henry
Michael Henry 2017-4-4
My be the problem with MATLAB version I am using. Can you please tell me which version you are using so I can use it.
Thanks again.
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Michael Henry
Michael Henry 2017-4-4
Thank you so much. I am able to run the code now with no errors after I downloaded the 2017a version.
Best Regards.
Star Strider
Star Strider 2017-4-4
My pleasure.
I am happy that it works for you.

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