## Monday Math 16

How might we find an exact, non-series formula in terms of α and β of the integral . Attempting to find the antiderivative of the integrand will show it cannot be done with elementary functions (it can be expressed in terms of a polylogarithm; see here). However, we do have a method that can do so: ‘differentiation under the integral sign.’

Define 
Now, let us take the derivative of this function. As the limits of the integral are constants, we see:

And thus
, with C some constant.
Now, we see
.

For a second example, let us consider
, 
As before, we find the derivative of this function:

Using a table of integrals (such as #21 here with a=1, p=1, and q=cosφ), we see that for 0≤θ≤π,

and thus

And this tells us 
Now, when , we have
, thus giving , and thus for , 

This method can also be used to find some definite integrals without parameters by adding one; for example, the integral

To use the method, we define:

Our original integral is thus 
Taking the derivative of this function,

Now, we see

Which is –αtimes the integrand we obtained for . Thus:

And thus  is a constant.
We see

And thus our original integral is also zero.

Similarly, one can find the integral from 0 to infinity of the sine cardinal function  by using this method upon

We see

And so we find
 for some constant.
Now, note that as , we see that for all x>0, we see
, and thus . As , we thus find , and so
, and our original integral is:
.