In this last integral for 2x substitute a; then

Let all the integrals in the right-hand member be brought to

π

the same limits, and 0; for this purpose in the second integral,

2

In the third integral, let x be replaced by π+x; then

In the fourth integral, let x be replaced by 2π-x; then

2

the value of which integral has been already determined in Art. 91. 95.] The following are cases in which the element-function contains a general function into which the subject-variable enters in a special combination, and in which the definite integral is simplified by transformation. The function denoted by ƒ in the following examples is finite and continuous for all employed values of its subject-variable.

In the second integral of the right-hand member for a sub

Ex. 2. By the same substitution it may be shewn that

x

(79)

dx

Ex.8. [*ƒ (2"+æ ̄") tan ̄1æ = S'ƒ (x* +

dx

f(x+x-") tan-1 x + [* f(x”+æ ̄") tan ̄1‚æ

In the second integral of the right-hand member let a be re

Ex. 4. By a similar substitution it may be shewn that

π

(81)

Ex. 5. Consider the integral ["a ƒ{sin æ, (cos a)2} da; then

x

since sin a and (cos x)2 are unaltered when x is replaced by π-x;

["xƒ{sinx, (cos.x)2} dx = −

Ο

−√ (π — x) ƒ {sin æ, (cos x)2} dx

π

= [ " (7—x) ƒ { sin x, (cos x)2} dæ ;

π

xf {sin x, (cos x)2} dx = ["ƒ {sin æ, (cos x)2} dæ. (82)

The following is an example of this theorem :

Many cases of transformation and of a consequent simplification of definite integrals will occur in the sequel: and the preceding are sufficient for illustrations of the

process.

SECTION 4.-On the Differentiation and Integration of a Definite Integral with respect to a Variable Parameter.

96.] As a definite integral in its most general form involves

the limits of integration, and any constants which may have been contained in the element-function, it may be considered as a function of these quantities, and treated accordingly. Thus, if u is a definite integral, of which x and x, are the limits, and a is. a parameter involved in the element-function, so that

n

u may be treated as a function of x„, x, a. And subject to the condition, which is always necessary, that the element-function is finite and continuous for all employed values of its subjectvariable, u may be treated as a continuous function of these three quantities, and differentiated and integrated accordingly. These processes we propose now to develop; and we shall have many applications of them; and from definite integrals, which have already been determined, others will be derived. In the most general case, o, and a may be considered as three independent variables; so that the total differential of u will consist of three partial differentials. There will also be particular cases where one, or two, of these three quantities will vary, the others being constant; these however may be treated as special forms of the general case.

Let the right-hand member of (84) be written at length; then u = (x1—x ̧) F′(α, X ̧) + (X2—X1) F′(α, x1) + .....

(1) Let x be increased by an increment xn+1-n, which is dr; then the corresponding increment of u is (x+1 — Xn) F′ (α, X„) ; so that for the partial differential of u due to the variation of x, we have

du

= F(a,xn);

(86)

X1-X0

(2) Let be increased by its increment a1-x, which is da。; so that the range of integration commences at 1; and u is diminished by (1—x) F′(a, x); consequently

(du)

= F(a,x);

dxo

(87)

Hereby we have the two partial differentials of the definite integral with respect to the limits.

(3) Let a vary; then, from (85),

du

da

d

= (x, − x ̧) — _ x′(a,x ̧) + (x2 −x ̧) — ¥' (a, x1) + ...

xn

d da

= ["d2 v(a, x) dx;

da

da

Con

which gives the partial differential of u with respect to a. sequently for the total differential of u we have

Hence, if x, xo, a are all functions of an independent variable t,

The process by which this equation has been found is commonly called differentiation under the sign of integration with respect to a variable parameter. Leibnitz has called it Differentiatio de Curvá in Curvam. The meaning of this remark will be plain from the following geometrical interpretation of (91).

Let Po PP, see Fig. 46, be the curve whose equation is y = F(a, x); let oмx, Oм,x,, then, as explained in Art. 3,

the area Mo Mn Pn Po

=

Let the parameter a vary, and first let the element-function alone vary, and let the new position of the curve which is due to the variation of a be LP',NP',, so that the area becomes increased by the quadrilateral LP PN: therefore

next let the limits vary, so that by the change of a, oм, becomes Oмo, and Oм, becomes oм'; and therefore the area is increased by PM, M',N' and diminished by PMML', which are respectively represented by F'(a,x,) dx, and F'(a,x,) dx,. But when all these variations are simultaneous, the definite integral expresses the area Po M' M', P', instead of PM, M, P, the two quadrilaterals LL', N N' being omitted, because they are infinitesimals of a higher