Here’s another definite integral problem: find a general formula for , with n an integer greater than 1.

First, let . Then , so , and our integral becomes:

.

Now, we perform the substitution , giving:

.

We should recoginze this last integral as a beta function:

, so we have:

.

Now, recall the Euler reflection formula . We see that this applies to the above with , giving us:

.

This is easily confirmed for by performing the original integral (arctangent). By factoring and , and using partial fractions, one can confirm and .

We also see that as expected by the fact that

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Tags: Beta Function, Euler Reflection Formula, Integral, Math, Monday Math

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June 15, 2009 at 12:22 am |

[…] . How does this relate to our integral? One may recall that . Now, . Thus, . In fact, . (See here for a similar use of the beta function and Euler reflection […]