. Applied calculus; principles and applications . +C= -itanh-i- + C. (x2<a2)2a ^a-\-x a a ^ ^ 190 INTEGRAL CALCULUS The first or second of these results is used according — aor a — a; is XIX, let Vx -f a2 = z-x; or z = x + Vx + a% (1) /. a^ = z^ — 2 {a?) = 0 = 2zdz-2xdz- 2zdx;{z — x)dz = zdx; dz dx dx z z-x Vx2 + a2/. r^L== ff = log. + C = log(x + ViH^) + C, or sinh-i - + C. (XIX) aFor XX, similarly, on letting Vx^ — a^ = z — x, I ^^ =log(x+Vx^-a^) + C, or cosh-i- + C. (XX) The logarithmic form of cosh-i - is log (^ r ^ x — a \ ^ but its derivative or differential is t
. Applied calculus; principles and applications . +C= -itanh-i- + C. (x2<a2)2a ^a-\-x a a ^ ^ 190 INTEGRAL CALCULUS The first or second of these results is used according — aor a — a; is XIX, let Vx -f a2 = z-x; or z = x + Vx + a% (1) /. a^ = z^ — 2 {a?) = 0 = 2zdz-2xdz- 2zdx;{z — x)dz = zdx; dz dx dx z z-x Vx2 + a2/. r^L== ff = log. + C = log(x + ViH^) + C, or sinh-i - + C. (XIX) aFor XX, similarly, on letting Vx^ — a^ = z — x, I ^^ =log(x+Vx^-a^) + C, or cosh-i- + C. (XX) The logarithmic form of cosh-i - is log (^ r ^ x — a \ ^ but its derivative or differential is the same as that of log{x + Va:^ — of), the constant a disappearing in the differ-edentiation; and so too with the sinh~i-. (See Art. 66.) a 121. Derivation of Formulas XV, XVIII, XXI, and XXII. — These formulas are merely the reverse of the differentialforms given in Examples 1 and 2, Exercise VI. They may be derived from the forms for the inverse trigo-nometric functions of a;. Thus:/f dxdx 1 I a _ 1 / ^ Va/ _ 1 . -\^ \ri smce. DERIVATION OF FORMULAS XV, XVIII, XXI, AND XXII 191 Since tan-i -=l- cot-i -, d f tan-i -] = d(- cofi -Va 2 a^ \ aj \ a) Hence /; ^^ =-tan-i-+C, or -icot-i- + C. (XV) 0? -\- x^ a a a a In the same way the second forms follow for formulas XVIII,XXI, and XXII. The standard forms are given in terms of -, because they are of more use than those in terms of x; the latter, beingspecial cases where a = 1, are often given as the standardforms. Integrals may be obtained by reduction to eitherform. EXERCISE XXin. 1 C dx _ 1 f d {ex) _ 1 ex ^ J 62 + c2a;2 c J 62 + {exY he 6 ^ ^* he 0 = c-/r^^^ = 2i-S^ + ^ (^^^^>^^) be 0 r dx r dx _ 1 ,„^-i3; + 3 ^ Ja:2 + 6x + 5 J (x + 3)2 - 4 4 ^ (a; + 3) + 2 ^ - r x^dx 1, x^ — 1 , ^ ^ r dx „ r xdx 1 , _, ^2 o r dx 192 INTEGRAL CALCULUS 10 r ^^ = 2 r ^^^^ J aa;2 4- 6a; + c J (2 ax + 6)2 + 4 ac - 62 tan ^ - — + C (4 ac > IP-) y 4 ac - 62 V 4 ac - 62 1 , 2ax + 6- V62-4ac , „ ,. ^ ,„,log + C. (4 ac
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