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Solving an Improper Integral

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Ahmad El Kari
Ahmad El Kari on 8 Apr 2020
Answered: John D'Errico on 21 Mar 2021
I've tried to solve the following integral as such:
syms x;
f = 1/(sqrt(exp(x)-x))
A = int(f, 1, inf)
and the answer was the function itself:
int(1/(exp(x) - x)^(1/2), x, 1, Inf)
Did I make an error because why is the answer not coming out?
Thank you.
darova on 8 Apr 2020
Maybe it's too complicated for symbolic calculations. What about numerical solution?
syms x;
f = 1/(sqrt(exp(x)-x))
F = matlabFunction(f);
ans =
ans =
doesn't look like truth

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Answers (2)

Deepak Meena
Deepak Meena on 21 Mar 2021
You did not make any mistake, To evaluate the A use vpa fucntion :
>> vpa(A)
ans =

John D'Errico
John D'Errico on 21 Mar 2021
Think of it like this. Do ALL problems you might pose to a computer have a solution? If so, then I might try
solve('Peace in the middle East')
To no surprise, MATLAB would have a fit if I try it, generating some randomly useless error message.
But many (even most) problems you might pose in mathematics actually have no analytical solution. It is frighteningly easy to pose such a problem. In fact, it looks like you did exactly that.
One solution, and a reason why there are numerical analysis courses taught at many schools, is to learn to use numerical methods to solve intractable problems, where no simple algebraic solution is available. Often numerical analysis is valuable even for problems where a solution exists in theory, yet it is numerically difficult to work with. But this is a probem of the first sort, where no analytical solution apparently exists. At least neither of MATLAB or Wolfram Alpha are able to offer one.
In MATLAB, you can then often tell MATLAB to look for a numerical solution. You can use vpa for that, or you can use vpaintegral.
syms x;
f = 1/(sqrt(exp(x)-x));
ans = 
But this will work
ans = 
Or you might do it as
ans = 


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