Parametric Differentiation – Examples

James Nearing in the book Mathematical Tools for Physics (Dover Books on Physics) shows a technique to avoid integration by parts. This is done by using Differentiation and Parameters for Integrals. I found this technique to be an interesting method for obtaining results to definite integrals. So here I play with the technique.

So here is the Technique as outlined in the book from section 1.2 Parametric Differentiation.

We want to do the definite integral

We do this by making a parameter attached to x.

In this case we will integrate:

If you take the derivative of this equation with respect to the parameter something nice happens.

If we repeat this and take the derivative again

And repeat

And repeat, and repeat, and repeat. This pattern continues.

And this means the nth derivative gives us

If we replace the parameter with 1 we get the result that we expect

So this is the example given in the section. There are two Exercises in the book that explore it. I give some more examples of this technique here as I found it quite interesting and relatively easy if you keep track of your signs.

So we start with Exercise 2

We parametrize this as follows

And we take the first derivative

this gives us

And we take the second derivative

This gives us

We set and collect terms to get

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