# integral binomial differential

• December 13th 2011, 05:33 AM
hurz
integral binomial differential
Hiow to evaluate
$\int_0^1{dx \frac{d^{2l-1}}{dx^{2l-1}} $(x^2-1)^{2l}$$
Using the binomial theorem
$\int_0^1{dx \frac{d^{2l-1}}{dx^{2l-1}} $\sum_{k=0}^{2l} \frac{(2l)!}{k!(2l-k)!} x^{4l-2k} (-1)^k$$

l is a non negative integer.

I don't know how to proceed. Is there another way?
Thanks.
• December 14th 2011, 03:50 AM
jens
Re: integral binomial differential
Move the differential operator into the sum. You'll need to evaluate

$\frac{d^{2l-1}}{dx^{2l-1}} x^{4l-2k}$

which you can easily do by induction. Some terms of the sum may vanish while doing this, I didn't check. Finally, swap the sum and integral. Evaluate the integral. And you should be done.