Tabular Integration by Parts
Calculus: Single Variable 5th edition, pages 348-351. (Note: The textbook does not
exactly elaborate on the following method, these are pages under the title “Using the
Table of Interals”. Integration by Parts can be found on pages 342-345.)
In Tabular Integration, let us assume an integral of the form
In order to arrive at an answer without dealing with tiresome algebra and
arithmetic, we can use the tabular integration method.
In using this method, we must assume, however, that:

The
function is a function that can be continuously derived until the
derivative is 0.

The
function is a function that can be continuously integrated easily.
We then make a table with two columns. One side has the derivatives of
and
the other side has the integrals of
. To each of the derivatives in the first
column, however, we alternate the sign. We then form terms by multiplying each
entry in the derivative column with the entry in the integral column that is just
below it. The answer, or antiderivative, is the sum of these terms. We can then
arrive at a general table, or general set of rules:
Column 1, the derivatives of
Thus:
Column 2, the integrals of
Example 1:
Derivatives of
Integrals of
Answer:
Example 2:
Derivatives of
Answer:
Integrals of
This method of integration can be used to find a variety of integrals, and it is the
method with which many basic functions and products of
,
, and
are arrived at. The following URL is a pdf file of a short table of indefinite integrals,
found on the back cover of our class textbook, Calculus: Single Variable, 5th edition
as well:
http://math.ucalgary.ca/files/math/courses/W04/MATH253/lec3/changs_indefini
te_integrals.pdf
Other helpful links:
http://www.youtube.com/watch?v=L2_JCyMfMzA
http://www.math.binghamton.edu/grads/reff/m222/LIATEandTABULAR.pdf