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. 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