What is integration by parts and why does it work?
Integration by parts is based on a formula:
It is a method to change one integral to another which may be easier to calculate.
This method is based on the formula for the derivative of ptroduct of 2 functions:
Now if we integrate both sides of the equation we get:
As an example I will calculate
In the example we changed an integral of
It works because of the product rule for derivatives and because integration is the inverse of differentiation.
The product rule tells us that
Using differentials, this says that:
more conveniently written
Now, Integrate both sides:
By the inverse relationship of differentiation and integration, we can simplify the left side to get:
And finally, subtracting
Rather than using the dufferential notation shown above, others prefer to use:
Still others write
(Or something like that. I've seen the last one rarely.)