1

–

x

dy

––

dx

1

–

y

improper integral 259

∂F

––

∂y

√1 – x2

√1 – x2

1

–

x

2

l

implicitly. This process is called logarithmic differentia-

tion. For example, to differentiate y= xx, write ln(y) =

ln(x)x= xln(x).Then ⋅= ln(x) + x ⋅, and so,

.

implicit function A formula of the form F(x,y) = 0 is

said to define the variable yimplicitly as a function of

x. For example, the equation xy = 1 implicitly defines

the dependent variable yas y= . However, matters

can be confusing, for a single equation may define yas

several possible functions of x(for example, the equa-

tion x2+ y2= 1 suggests that either y= or

y= – ), and in some instances, it might not even

be possible to solve for an explicit formula for y(for

example, it is not at all clear that x2y2+ ycos(xy) = 1

does indeed give a formula for y).

Mathematicians have proved that as long as the

function F(x,y) has continuous

PARTIAL DERIVATIVE

s,

and that there is some point (a,b) for which F(a,b) = 0

and ≠0 at (a,b),then the equation F(x,y) = 0 does

indeed define yas a function of x, at least for values of

xwithin a small neighborhood of x= a. Moreover, the

process of

IMPLICIT DIFFERENTIATION

is valid for these

values of x. This result is known as the implicit func-

tion theorem.

improper integral (infinite integral, unrestricted inte-

gral) In the theory of

INTEGRAL CALCULUS

it is per-

missible to extend the notion of a definite integral

∫b

af(x)dx to include functions f(x) that become infinite

in the range under consideration, [a,b], or, alternatively,

to consider integration over an infinite range: [a,∞),

(–∞,b], or even (–∞,∞). Such integrals are called

improper integrals. One deals with such integrals by

restricting the range of integration and taking the limit

as the interval expands to the required size.

Consider, for example, the integral . The

integrand becomes unbounded as xapproaches

the value zero, and so, in the normal way, the integral

is not well defined. However, the function is

bounded on the interval [L,1] for any 0 < L< 1, and

the integral is valid. We have:

= 2 – 2√

–

L. The improper integral under consideration

is then defined to be the

LIMIT

as L→0, from

above, of these values:

.

In general, if an integrand f(x) is infinite at an end

point a, then the improper integral ∫b

af(x)dx is defined

as the limit:

∫b

af(x)dx = limL→a+∫b

Lf(x)dx

or, if infinite at the end point b, then the improper inte-

gral is defined as the limit:

∫b

af(x)dx = limL→b–∫L

af(x)dx

If the limit exists, then the improper integral is said to

converge. (Thus, for example, is a convergent

improper integral.) If the limit does not exist, then the

improper integral is said to diverge. (One can check,

for example, that diverges.)

To illustrate the second type of improper integral,

consider, for example, the quantity . This is to

be interpreted as the limit:

which can be readily computed:

. (We

interpret this as saying that the total area under the

curve y= to the right of x= 1 is 1.)

In general, improper integrals of this type are com-

puted as follows:

∫∞

af(x)dx = limL→∞ ∫L

af(x)dx

∫b

–∞f(x)dx = limL→–∞∫b

Lf(x)dx

∫∞

–∞f(x)dx = limL→∞ ∫L

–Lf(x)dx

1

–

x2