Integration by Parts
The message of the fundamental theorem of calculus is that integration and differentiation are
inverse processes. Whenever a statement is made in one of these subjects there is a
corresponding statement in the other. Not all of these corresponding statements are useful: for
example, the statement in integration theory corresponding to the quotient rule in differentiation
is not useful. However many are useful. The chain rule corresponds to the method of substitution
in integration. The product rule corresponds to integration by parts, which is the topic of this
note.
To start recall the product formula:
Now taking the integral of this formula gives:
This is the basic integration by parts formula. As you can guess it is not usually left in this form.
Here are some of the equivalent forms:
Some explanation of the notation is needed. The equation (1) is just a rearrangement of the basic
formula. In equation (2) we have used the notation that functions denoted by lower case letters
are the derivatives of the corresponding upper case ones, so
F'(x)=f(x) and G'(x)=g(x)
. In
equation (3) we have used
u(x)=f(x)
and
v(x)=g(x)
, then written the derivative terms as
differentials:
du=f'(x)dx
and
dv=g'(x)dx
. Thus these three equations are actually the same one in
three different notations. Notice that in all three versions there is an integral on both sides of the
equality sign. Thus by implication there is a constant of integration on both sides, which has
absorbed the constant of integration shown in the basic formula.