the sign of the last integral having been changed by the reversal of the limits.

As this process admits of extension to any one or more values of a beyond x-xo, provided that F'(x) does not become infinite or discontinuous within the range of integration, it follows that the theorem enunciated at the beginning of Art. 85 may be enlarged so as to include all values of x, which by algebraical addition give x-xo, over which range the sum of the elementfunctions is to be taken.

88.] Hence also may be deduced two theorems of great use in the evaluation of definite integrals.

Firstly, if is the arithmetic mean between x and x the limits of integration, and if F'(x) has the same value and the same sign at equal distances from έ on either side of it, that is, if r′(§—x) = F′(+x); then the series, the sums of which are

*}

denoted by fr F′(x) dæ and [**r′(x) da, will consist of terms which

are, term by term, equal to each other; and consequently

If έ, the arithmetic mean of the two limits, is zero, so that xox, and F'(x) = -r'(x), then

=

ท

a

Ex. 3. [^(a2 —x2)3 dx = 2[* (a2—x2)3 dx =

Ex. 4.

2.

x2ne-a2x2 dx = 2 2 [* x2ne-a2x2 dx.

Also generally, if R denotes a rational function,

πα

2

Secondly, if is an arithmetic mean between x, and x, the limits of integration, and if F(x) has the same value, but of different signs, at equal distances from έ on either side of it, that is, if F(+x) = −F′(§—x); then the series, the sums of

'હૃ

which are denoted by [*r′(x)dx and F(x)dx, will consist of

Το

terms which are, term by term, equal to each other, and of contrary signs; and consequently

so that the sum of the two definite integrals is zero. Hence

If 0, so that xx, and F'(x) = F(x), then

=

-

[**¥'′(x)dx = 0.

The following are examples of these theorems.

cos x dx +

π

cos x dx

(38)

(39)

Also, generally, if R denotes a rational function,

Ex. 4. [*n {sin æ, (cos x)3} cos x de_ = 0.

0

(40)

89.] Hitherto in evaluating [**r′(x) dæ, F'′(x) has been assumed

to be finite and continuous for all employed values of its subjectvariable; and our inquiry has been restricted to cases wherein this condition is fulfilled. Suppose however έ to be a value of x, within the range of integration, for which F′(x) becomes either infinite or discontinuous, the theory of a definite integral which is given in Art. 85 enables us to treat of this case. Let us divide the integral into two parts, as follows;

μ

(41)

Now the integrals in the right-hand member do not admit of treatment, because they have no determinate value according to the principles of the preceding Articles, inasmuch as the limits include values for which the element-function is infinite. Suppose however i to be a general symbol for an infinitesimal, and μ and v to be two arbitrary and undetermined constants; and let έ, which is the superior limit in the first integral of the right-hand member of (41), be replaced by έ-μi, and let έ which is the inferior limit in the second integral be replaced by §+vi, so that in neither integral does the range include that value of the subjectvariable, for which the element-function is infinite or discontinuous. Then if the definite integrals in the right-hand member of (41) are determined for these limits, μ, v, and i will generally enter into them; and if after integration O is substituted for i, the required definite integral will be found. Thus we have

If in the determination of a definite integral it is convenient to extend the range of integration beyond the original limits of integration, say to έ beyond the superior limit a,, and if & is a value for which the element-function is infinite or discontinuous; then the limits must be taken as follows;

ξμί

"F'(x) dx = [* ̃ “r'(x) dx + [* "r′(x) dx,

-vi

(43)

and zero must be substituted for i in the result. A similar formula will be required if έ is a value below the inferior limit.

ૐ

Similarly if r′(x) is infinite or discontinuous for many values of x, say for x1, x2,...,-1, lying within the range of integration; if i is the general symbol of an infinitesimal, and 1, 1, M2, V2 ... are positive and arbitrary constants, then

i being in the result replaced by zero.

The following are instances of the process.

Ex. 1. In

in f'da

-1

, the limits include zero, for which the element

function, x, x; we must consequently divide the integral into two parts, of which the limits will be -ui and −1, and 1 and vi: thus,

dx

L'da

=

-μidx
+

dx

1

which quantity is definite as to form, but indefinite as to value, because μ and are thus far undetermined quantities.

Ex. 2. In the integral (*

π

dx

a+b cos x'

if a is greater than b,

the denominator is always positive, and, see (27), Art. 67,

But if a is less than b, the denominator = 0, and changes sign

Let this value of x = a; so that the element-function = ∞, when x = a; consequently,

consequently, by (28), Art. 67, substituting in (47),

which =

0

sin (a + ") sin

8, when i = 0; but, on evaluation, we have

(48)

(49)

which is determinate in form, but undetermined in value, because μ and v are constants thus far arbitrary.

according as a is greater than, equal to, or less than b.

90.] The case in which one of the limits of a definite integral is ∞, either positive or negative, and the element-function corresponding to that limit = ∞, may be treated by a process similar to that of the preceding Article: for we may replace

1

by

+ and in the evaluated definite integral substitute O for i. μής Thus if F(x) = ± ∞ when x = ∞, either sign being taken,