206 Integrals
Thorough understanding of this particular example is of utmost impor-
tance because it involves an idea that is very often used to overcome a
difficulty. In (4.183) we have an integral of the form fu" dx which does
not have the formfun (du/dx) dx. However, du/dx is 5, a constant,
so we can insert the factor 5 in the integrand and compensate for the deed
by inserting the factor - before the integral.
To obtain the formula
(4.185) f
(^1) +x2
2x dx = log (1 + x2) + c,
we read the left side "integral of one over u dee u dee x dee x" and apply
(4 175). If the factor 2 had been missing from the integrand in (4.185),
it would have been necessary to insert the factor and compensate for the
deed. Thus
(4.186) f1 + x22x dx = 2 log (1 -I x2) + c.
x2dx = 2J 1 1
Our very modest table of integrals beginning with (4.171) does not
reveal the answer to the question whether there are any functions F(x)
for which the formulas(4.187) F'(x) =
1 -- x2' f1 -L x2
dx = F(x) + c
are valid. Many useful purposes are served by this table and the more
extensive one appearing opposite the back cover of this book, but one
who has solved several of the problems at the end of this section is ready
to recognize the fact that there exist much more elaborate tables of
integrals. The books of Buringtonf and Dwight$ are exceptionally use-
ful examples of books that give hundreds of integration formulas, tables
of values of functions, and other mathematical information. It is possi-
ble to proceed through our course without using tables other than those
on the back cover and facing page of this textbook. However, students
who contemplate following educational programs in which mathematics
appears are well advised to purchase one of these books (or perhaps
another more or less similar one recommended by teachers) and to spend
occasional moments (and sometimes hours) inspecting its organization
and studying its contents. Ability to understand and use the tables is
not inherited but can develop rapidly as more calculus is learned.
Experience shows that persons who have completed courses in calculust R. S. Burington, "Handbook of Mathematical Tables and Formulas," 3d ed., McGraw-
Hill Book Company, Inc., New York, 1948, 296 pages.
t Herbert Bristol Dwight, "Tables of Integrals and Other Mathematical Data," 4th ed.,
The Macmillan Company, New York, 1961, 336 pages.