Calculus: Analytic Geometry and Calculus, with Vectors

4.1 Indefinite integrals 207

refer to tables in books of tables in preference to tables in calculus text-

books. Teachers can be particularly helpful when they require that

their students purchase identical books of tables and make frequent

comments about use of the tables. Sometimes use of a book of tables is

permitted in tests and examinations where use of a calculus textbook is

forbidden.

Problems 4.19

1 Tell the meaning of ff(x) dx. Be prepared to give full information at

any time.

2 Show that, when x is properly restricted,

(a) f(2+3x+4x2)dx=2x+3222+433+c

2 2

(b) f 1-xdx+2+c

2

(c) f ( x

1

)dx=3(x+3)\Ix +c

(d) f x(1-x)dx=22- 3+c

3 Is the formula

fx3 dx = xfx2 dx (?)

true or false?

4 Brevity is sometimes but not always a virtue. It can be claimed that the

second formula in (4.171) would be much more easily understood and used if it

were written in the form

f [u(x)]"u'(x) dx=[n(+1 + c.

Think about this, and then write the other four formulas in terms of the Newton
notation for derivatives. Note that the last formula takes the form

f u'(x) dxU = log1U(X)1 + c.

Remark: We can abbreviate (4.171) to the form fun du = u"+'/(K + 1) + c, but
for present purposes we can hold the view that further abbreviation of (4.171) is
a step in the wrong direction. We need not be in a hurry to join the ranks of
gullible people who think that the du appearing in the symbol fu du is a number
because a correct result is obtained by pretending that du is "the" differential
u'(x) dx and writing

fu du = fu(x)u'(x) dx = 1''[u(x)]2 + c.

When we do not use the abbreviation, we do not need to worry about it.