Numerical integration over (0, x1)

I am trying to numerical integrate the function. I am not able to solve it. Can anyone help me?

Note: t is a constant over here and limits are from 0 to x1.

Thanks in advance.

Raj Patel.

[John D'Errico]

PLease don't insert a picture of code. If it were text, I could just copy your function into MALTAB to look at it. As it is, then I need to retype it, for no good reason. And while this is a simple case, why in the name of god and little green apples would you want to make it more difficult for someone too help you?

[John D'Errico]

Next, is t a known constant? Does it have some value? What is it? If not, then you cannot use integral to integrate a function with a symbolic parameter.

[Raj Patel]

Thank you for pointing that out John. I didn't realise it. But I have copied the code below.

fun = @(x)(1 ./ ((exp(0.014342 / (x * t)) - 1)) .* (1 / (x^4)));

q = integral(@(x) fun(x), 0, x1)

Thanks,

Raj Patel

[Raj Patel]

"t" is an unknown constant. What shall I use to integrate it if I cannot use integral?

[Raj Patel]

I tried using int(function) to solve, but I am still not getting an answer.

[Raj Patel]

https://www.mathworks.com/matlabcentral/answers/597415-numerical-integration-over-0-x1#comment_1018222

I tried using int(function) to solve, but I am still not getting an answer.

Answer 1

[John D'Errico]

If t is unknown, then you need to use symbolic tools to do the integration.

syms x t
K = 1/((exp(0.014342/(x * t)) - 1)) * (1/(x^4));

int(K,x,[0,1])

ans =
int(1/(x^4*(exp(2066900027383925/(144115188075855872*t*x)) - 1)), x, 0, 1)

MATLAB just returns the integral in the form it was given. I tried Wolfram Alpha, which also agrees it cannot find a solution. There is no assurance that anything you write down has a nice closed form solution.

You have some options. If you had some value for t, then we could write it as:

Kxt = @(x,t) 1./((exp(0.014342./(x * t)) - 1)) .* (1./(x.^4));
foft = @(t) integral(@(x) Kxt(x,t),0,1);

foft(3)
ans =
           22003287.386445
           
foft(1.2)
ans =
          1408175.40694754          

So while no analytical form is available, we can find a numerical solution. I could have used vpaintegral too.

Finally, you could probably write the problem as a series expansion, but then you would need to consider the domain of convergence, etc.

[Raj Patel]

Thank you John. I appreciate your effort and time.

Answer 2

[Alan Stevens]

Need to write fun as

fun = @(x) (1./(exp(0.014342./(x*t))-1).*1./x.^4);

[Raj Patel]

Thanks Alan.