44 INFINITESIMAL CALCULUS (2) If n be less than unity, the equations / & it j "A <" / -a^rr^ = T tan -2 can be readily established. (3) If a<ir, by a simple transformation it can be shown that -co ax_-ax (4) If + &<ir, we can prove the equation 2 cos cos 2 2 cosa+ cos b (f) To find the value of Assume dx __ 1+7/ 6 (6) In like manner /- 145. We now propose to consider some of the general methods of evaluating definite integrals. It is obvious that the value of the definite integral /& f(x)dx is independent of the variable x, and is a function of the limits a and b, as also of any constant parameters contained in the function /(#). We proceed to show that a definite integral may be differen tiated, and also integrated, with respect to any such parameter. Differentiation under the Sign of Integration. 146. Suppose the function f(x) to contain a constant parameter a ; i.e., let f(x) = (j>(x, a); then, denoting the definite integral by u, we have /b 4>(x, a)dx . Also, let the limits a and b be independent of a ; then, if Au de note the change in u arising from the change Aa in a, we get Ja ( ^ a + Aa ) ~ $( x > a ) } dx Aa Ja Aa Hence, passing to the limit, du _ r*> d(j>(x, a) , da Ja da This principle is called differentiation under the sign of inte gration, and, by aid of it, from any known integral a number of others can in general be determined by differentiation with respect to the constants contained in the integral. For example, if we differentiate the equation /"* 1 we get / Jo a and, by n successive differentiations, 1.2.3.. , . n Again, if the equation be differentiated with respect to a, b, c respectively, we have f ire -/ _ x (a + 2b,>: + cj:-)~ ~~ 2(ac-b~)2 f xdx irb - + 00 x-dx Hence A number of other definite integrals can be immediately deduced from these by successive differentiation. Again, since f - = -L_ , -/() (a + 2fcc + c# 2 )3 ha * where 7t= Vac + i, we get, by differentiation, In like manner, if the equation
- T dx
y<> a 2 cos 2 a; + 2 sin 2 : 2a/3 be differentiated with respect to o and j8 respectively, we get /" cos^xdx IT yo (a- c Hence, by addition, 2 sin 2 a;) 2 4 a/8 3 yo (a 2 From these other definite integrals can be readily found by further differentiation. 1 47. When the limits are functions of the parameter o, a de finite integral admits of differentiation in like manner. For, let A, Ab be the changes in the limits corresponding to the increment Aa in a, then < (x, a. + Aa) dx - I <j dx + a)dx (x, a + Aa)dx. JO: Hence, proceeding to the limit, we got d u - r - da d(f>(x, a) ~ ^-^ da db da v . . a) -. -- <b(a, a) -y- da. da Integration under the Sign of Integration. 1 48. We shall next consider the corresponding process called in tegration under the sign of integration. Suppose / <f>(.v, a}dx to be represented by u, then Ja I f b x, a)da kfa=/ <t>(r, a)dx = u , the same limits for a being taken in both integrals. Suppose oj and o to represent the limiting values of a, then the preceding result may be written /b r~ /" a i n /* a i r /~b ~ / $(x, a)da b:=/ / <}>(x, a)d.r Ida ; ( AH, j ya uy"
