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.