We can find the 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, this gives , as expected.

For non-zero *k*, we integrate with respect to *a*, and find

.

Again, we note that as , we see that for all *x*>0, , and thus .

For *k*>0, we have , giving , and so .

For *k*<0, we instead have , giving , and thus .

Thus .

Now, to our original integral .

Define , for *p*≥0

Then

,

and now we know, from our previous work, that for *p*>0, this is . Integrating, , and as , we see , and so our integral is

.

2. Difficult: Trig and Double Integral.

We can use the trigonometric identity to rewrite our integral:

.

Substituting , we have

Now, here is the hard part: we can recognize that . Thus, , and our integral can be rewritten as the double integral

.

We can reverse order of integration to solve this:

3. Simple (but needing some background): Fourier Transform.

Defining the sine cardinal as . It is known that the Fourier transform transforms between the sinc function and the rectangle function: namely, , where the function Π(*x*) is the rectangle function

The convolution theorem tells us that:

,

where the * represents convolution. The convolution of the rectangle function with itself is the triangle function:

.

And so ,

Writing this out with the definition of the Fourier transform, we have:

.

Plugging in *f*=0, and using Λ(0)=1,

.

The sinc function is even, so we have

Using the subsistution *x*=π*t*, we have

QED.

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Tags: Integration, Math, Monday Math

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