How can I estimate specific heat integral?

Question

Please help where can I find the procedure or numerical values for specific heat integral

integral {from 0 to x } of x^4exp(x)/[(exp(x)-1)^2]

Answer 1

Many thanks for the answer.

I know the estimation for small x i.e. x<<1, but I can not use it.

Unfortunately, the integral which I need to calculate is definite i.e. from 0 to s and can not be converted to indefinite :(

I am looking for the procedure or approximation by a series or tables of this integral. In the end I need numerical values.

Answer 2

as I recall these integrals from (what we used to call) solid state, I believe that the values of x are small compared to one; if this is a correct assumption, then just expand the exponentials in power series:

exp[x]=1+x+x^2/2! +...

the denom becomes:

(exp[x]-1)^2=(1+x-1)^2 = 1 (using only first order terms which is ok if x<<1

so then you have a simple integral:

Int[x^4(1+x) dx]

if you cannot make the assumption of a small value of x, then you have a very complicated integral requiring some very advanced functions (like polylogs)

let me know if you can if x<<1 is permissible in your case

note added in editing...thank you for your additional comment about needing this for large x...as you know, this integral is pretty hairy, and I would attack it using contour integration techiques from complex analysis...but...since I like to do things as simply as possible, I plotted the entire function over the range (0,10) and found that for x>4, we can easily use the large x approximation; that is, exp(x)>>1, so the denom just becomes exp(2x), and the integrand becomes x^4 exp(-x) which integrates easily (by parts).

so, for x<1 you can use the small x approx, for x>4 you can use the large x approx, and for values between 1 and 4 I would suggest using numerical techniques

(the analytic integration of this function involves polylogs, whose evaluation involves zeta functions...and I am not sure how deeply into applied math you want to get)

hope this helps