CAPUT VI,

-57

V

350, Corollarium 1.

Si datas exponens differentialis ,’Xex solam fun&ionem irrationa-lem /(a + cx 2 ), quomodocunque cum variabili x et ejus functionibusrationalibus connexam, comple&atur; valores, pro quibus, loco x, ex,yf (a + cx 2 ) fubftitutis, ille formam rationalem induet, erunt, ob (349- §•)pro b=0, sequentes.

I. Pro coefficiente pofitivo quadrati x 2 .

V^Ca + cx 2 )

z 2 + i

ex:

2 z

(z 2 + a) e z *2Z 2 / C

1 2 z/C ’

pro z^x^c-f v^( a + cx 2 }.

II. Pro coefficiente negativo quadrati x 2 .

2z/a _ (k — z 2 )y^a

yf C*—cx 2 ):

S X

I+Z 2 '

— 4z £z/a

=fT+^vV Pr ° Z:

(1 + z 2 ) /c, \f (~a— cx 2 )

'\f a + x^c

351. Corollarium 2.

Per (350. §.) determinari poliunt sequentia memorabilia integralia.Polito z = x +^(« + x 2 ).

J(x + /(.«■ + x 2 )) n e X

+ -f^ n “ 2{Z =

7 n 4- *

+

.(X+V^ + X 2 )) n + ’

+

2 (n -f 1) r 2 (n — I)«(x-f/fe-fx 2 )) n ~ 2

2(n —1>

2(n + i)

Unde pro n =— 1 obtinetur per (246. §.)

*£ + /(« + x 2 ) =l:I(x+ ^ Ca + x2) )— 4 (x + /(«-l-x 2 )> +C **7(ex + c/(a+ x 2 )) (x + /'(* + x 2 ))" ex —

— 4 C e + 0^ n + I ez -s- —Jz n ~ z ez-\-— (c—e} v /z"" I £Z nr

2 4

(e-j-c) (x-f-/ - (a:-fx 2 )) n + 2 «c(x-f-/"(a-f-x 2 ))® , a^c-e^fx/'fa-f x z )) n “ 2 —4 C n + 2} 2 n 4 (n — 2)

U 3

Hinc