My students claimed they were doing a calculation that required

.

I’m not sure what physical situation brought up such a question, but we can find the answer anyway. Let’s kill infinity birds with one stone by evaluating

.

and treating my students’ problem as a special case.

First define the gamma function by

.

I have never understood why involves as the power of , rather than just . It makes even less sense when you consider for natural numbers .

In the definition of the gamma function, make the substitution

We can choose whatever we want for , as long as we think we can find . So let’s turn this into the original problem by substituting

Putting it all together:

.

So we understand these seemingly-more-complicated definite integrals equally as well as we understand the gamma function.

For the special case my students were interested in, which has , the integral goes from to , so we need to multiply by two to get

.

A computer tells me this evaluates to about 1.8.

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Tags: calculus, definite integral, gamma function

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