Monday Math 45

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
, x,y>0.

One family of integrals this can solve is , n a positive integer.
Note that .
Now, .
Thus .
We know ,
so

Using recurrence relation , we see , and thus
.

Now consider . Here, we examine the mixed second partial derivative of the beta function:
.
Thus

And we found

Now, ,
, , and
.
Plugging in these, we see

(See #7 here).