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*=cos*φ*), 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 –*α*times 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:

.

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Tags: Calculus, Integration, Math, Monday Math

This entry was posted on April 21, 2008 at 1:27 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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July 28, 2008 at 12:04 am |

[…] definite integral in three different ways: 1. Long: Differentiation under the integral sign. In my previous post on this topic, I showed that . We can use a similar method on , namely, defining , and then finding . For k=0, […]