MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
‘21 2 
Ex. 7'fdx*/ X J rX n. 
This may be put in the form 
f dx (a + hx > -v{ (a+ L + + ..■■)» • 
+ Xn 
and putting 
1 + X r 
= V , 
(a -f- b x + . . . .) s 
the reasoning is the same nearly as in the preceding case. The same principles are 
applicable to the more general integral J* dx ^/ i _j_ x n , m being a whole number. 
These solutions give the algebraic sum of n integrals of the proposed form. But 
this number n may be reduced by various methods to a lower number, which is 
the minimum that the problem admits of: ex. gr. the lowest number of integrals of 
the form J* d x \/ \ -f^ 4 which have an algebraic sum is two ; of the form J* dx V\ #5 
is three ; of the form J* d Xy/T+x™ is likewise three, &c. &c., which subject I shall 
treat of in a subsequent section. 
_ P d x 
Ex ■ 8 -JyW : =T 
First solution. Put xr* = t, and the integral becomes 
%t~*dt _ i 
d t 
Put t 3 — t 2 — v, or t 3 — t 2 — v = 0. The three roots of this equation answer the 
problem. 
Z* 1 S dt P 
the sum of three integrals = J ' —J ® — const. 
(because S t = 1, being the coefficient of the second term of the equation t 3 — t 2 — 
v = 0 taken negatively, whence S d t = 0). 
Second solution. Put x? = t 2 , and the integral becomes 
Put t 3 — t = v, 
§t~i dt _ 2 dt. 
1 _ 3 
: . t 3 — t — v = 0, 
and the roots of this equation answer the problem. 
.•. the sum of three integrals = J * = J ' 0 — const. 
(because S t = 0 in the equation t 3 — t — v — 0 .\ S dt — 0), and the sum of the 
integrals therefore reduces itself in this case also to a constant. 
It will probably be satisfactory to the reader to see some one of these results veri- 
fied by arithmetical computation. Let us therefore select this last example for that 
purpose. 
