# Thread: [Help] integrate a nasty function

1. ## [Help] integrate a nasty function

Dear all,

Could anyone help me integrate the expression below? Its form is a bit nasty, containing both exponential and power functions.

F(X) = X^n/[1+exp(X)], n is a positive real number, does not have to be an integer.

Really urgent to get answer. Thanks!

Sam

2. Originally Posted by yjwang05
Dear all,

Could anyone help me integrate the expression below? Its form is a bit nasty, containing both exponential and power functions.

F(X) = X^n/[1+exp(X)], n is a positive real number, does not have to be an integer.

Really urgent to get answer. Thanks!

Sam
There is no solution to this in terms of elementary functions. I'd suggest a power series expansion, then integrate term by term.

-Dan

3. Originally Posted by yjwang05
Dear all,

Could anyone help me integrate the expression below? Its form is a bit nasty, containing both exponential and power functions.

F(X) = X^n/[1+exp(X)], n is a positive real number, does not have to be an integer.

Really urgent to get answer. Thanks!

Sam
$\displaystyle F(x)=\frac{x^n}{1+e^x}$ may not have an elementary solution (doesn't look so outright, so I am trusting topsquark on this one), but

$\displaystyle F(x)=x^n(1+e^x)$ does have a solution when integrated

4. If the limits are 0 and $\displaystyle \infty$ then the integral can be expressed in terms of $\displaystyle \zeta(n)$ by writing $\displaystyle \frac{1}{1 + e^{x}}$ as $\displaystyle e^{-x} (1 + e^{-x})^{-1}$ and using the binomial expansion.

If the limits are t and -t, then the integral will clearly be zero if n is odd; if n is even, it can be found by making the substitution x = -y.