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).

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Tags: Beta Function, Digamma Function, Gamma Function, Integration, Math, Monday Math, Partial Derivatives, Polygamma Function

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