Integration by Parts (Partial integration)
Integration by parts (or partial integration) is one of the most useful basic calculus operations. It is
a generalization of the fundamental theorem of calculus for the case of the product of two functions.
Fundamental theorem of Calculus
The classical definition of the definite integral of a function on the closed interval
Integration by parts
The second important result from calculus is the form of the Fundamental Theorem known as integration by parts
or partial integration. If we consider two differentiable functions , then
we may write by the product rule
Integrating both sides from to yields
which may be written more simply as
Recognizing that the term on the left is an exact differential, we may write
This expression may be rearranged to yield the familiar form of integration by parts
A generalization of integration by parts to higher dimensions is the Divergence Theorem also known
as Gauss's Law, particularly when applied in electromagnetism.
- ↑ Greenspan, Harvey Philip, and David J. Benney. Calculus: an introduction to applied mathematics. H, P. GREENSPAN, 1997.