As we noted previously, the derivatives of the beta function can be used in solving various integration problems.
First, we have the beta function definition
One family of integrals this can solve is , n a positive integer.
Note that .
We know ,
Using recurrence relation , we see , and thus
Now consider . Here, we examine the mixed second partial derivative of the beta function:
And we found
, , and
Plugging in these, we see
(See #7 here).