# Solving an Improper Integral

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Ahmad El Kari on 8 Apr 2020
Answered: John D'Errico on 21 Mar 2021
Hello,
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.
##### 2 CommentsShowHide 1 older comment
Ameer Hamza on 8 Apr 2020

Deepak Meena on 21 Mar 2021
Hi,
You did not make any mistake, To evaluate the A use vpa fucntion :
>> vpa(A)
ans =
1.3725437567734794327982409932998

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));
int(f,1,inf)
ans =
But this will work
vpa(int(f,1,inf))
ans =
1.3725437567734794327982409932998
Or you might do it as
vpaintegral(f,1,inf)
ans =
1.37254