INFINITESIMAL CALCULUS 45 or, adopting the usual notation, ,i , a)dadx-- (x, a)dxda . Such expressions are called double integrals, and the result just es tablished is equivalent to the statement that in a double integration, when the limits are independent one of the other, we may effect the integrations in either order without altering the result. It is easily seen that the preceding statement does not hold if a) a) become infinite within the assigned limits of integration. By aid of this principle, from a definite integral involving constant parameters we can often obtain others by the method of integration with respect to one of its parameters. (1) For example, if the integral / c ax cos bx dx = - p, a 2 + b* be integrated with respect to a between the limits a and /3, we get f Jo (2) If the same equation be integrated with respect to b between the limits a and 0, we get Jo On making a = in this, it becomes /* sin ax x provided a is positive. (3) Again, if the definite integral ( 133) /TT dx TT 1 + K COS X -S/l _ K l be integrated with respect to K, between the limits and sin a, we get /*" log (1 + sin a cos x) dx = ?ra . COS X (4) Next, if the equation /I i ~n+l be integrated with respect to n between the limits a and /3, we get r l ^-x a _ 1 + 3 /O log~T " i0 l + a- (5) To find the value of the integral /-co / -* 2 ,J u=/ e dx . Jo If zx be substituted for x, we have, since the value of the integral plainly remains unaltered, Hence, integrating with respect to z between the limits and oo , we have /-JO /- /-C / / - 2 2 (l+;r 2 ) , 7 / / c zdzdx = u/ o Jo Jo Consequently u = ^fv : I.e.,/ c a dx = ^/Tr . Jo 149. In many cases an unknown integral can be reduced to an elementary form by differentiation under the sign of integration. For example, let u= I - dx ; Jo x/l - x 2 then d " fl * 1 = , (Ex.5, 113). No constant is added, since u = when = 0. 150. A modification of this method of determining definite integrals is founded on the transformation of a simple integral into a double integral, and the inversion of the order of integration. (1) For example, when x is positive, M-C have I sinaxdx/ c~ x dy o Jo Jo x = fC Ay fC e " z * siu ax dx = y " Tt^+7 5 = ^ ( 148)- (2) Next, to find the value of i + b sin 6 dd
a-o sin 6 ) sin Hero, from the elementary equation i a + b sin 9 f 1 2ab sin rte re get lotn - = / V a - b sin 9 J Jo a- - b -x~ sin -0 o 7 f 2 f 1 dxde . . it = 2a6 / / yo Jo a ~ v~ x " sin-0 Hence, inverting the order of integration, we get ( 133) 7 r 1 dx . y U=-irt)l =TT SHI - 1 ! Jo /a 2 - b -x 2 Similarly we get re- /r / a + b sin ( ir^ .-
a - b sin
Ex. 3. Again, by aid of the equation 1 1 b n f 1 ff&sin 0dx tan- 1 sin =/
a /Jo a + b~x- sin-0
it is readily seen that ,4- 1 Han- n n 7 "* sin 6 sin 6 dQ = a 2 151. Lagrangc s Theorem. That Lagrange s series ( 56) can be established by the integral calculus, and its remainder after any number of terms exhibited in the form of a definite integral, was shown by M. Popoff (Comptes rendus, 1861). His demonstration has been transformed into a simple shape by M. Zolotarelf, in the following manner. Let z = x + y<p(z], and suppose the definite integral ~ If we malce n = l, we have In like manner, if n = 2, we get Consequently Again and so on. Hence we deduce- finally
