_ C A P V T IV.

*43

Problema 22.

131. Formulae dijserentialis dy—x nx dx intc*grale inuesiigare , ac per seriem infinitam exprimere.

Solutio.

Commodius hoc praestari nequit, quam vt for-mula exponentialis ^ n3 ‘ in seriem infinitam conver-tatur , quae est

. 1 , n* x’U* )* . n* x 1 ' 1 x ) 3 , n*x 4 ( l x i*

x nx -i-hnxlx + —171—ri———+TT1.7- + ctc.qua per dx multiplicata , et singulis terminis ia-tegratis, erit

sdx~x

sxdxlx — x* [--—£)

fx'dx[lxf ^

sx*dx{Ix)*~x*(

ij[lx)* t(lx)* ■ 4< 3 (l « )' t. s-ilx , 4-r-r.r^f ~ s 1 ' i 3 ~

etc.

t

'+~s» r

Quare fi haec series substituantur , et fecundum po-testates ipsius /or disponantur , integrale quaesitumexprimetur per has innumerabiles series infinitas:

r nx j _ _ 1 t - nx , n 7 x* n 3 x 3

jr—sx nx dx~-+-X(i- ? 4—p™p--

nx*lx ( 1

n*x*

-etc)

4-

»X

JT2X

n*x*{Zxl 2

5 *

n*x* .

j» etc.)

4 -

n*x*

. n , x*(Zx) 5 f 1 nt . n*x*

4- ~ ^ 4- ys —4

etc.

«♦ 4-“— etc.)

n*x* \

Hir— etc.)

quod