194 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
This gives 
or 
= x \/ 1 + x n , 
x n T " -j- x 2 — r 2 = 0, 
an equation, the product of whose roots must be + r 2 . But by the hypothesis this 
product is always represented by r. Whence it follows that no solution of the re- 
quired problem is effected by the supposition \/ 1 + x n — And at the same time 
we see that in order for any supposition to be successful, it is necessary that the re- 
sulting equation, arranged according to the powers of x, should have r for its last 
term. 
Now let us remark, that if S J'dy x . d x has an algebraic sum, then S J' x -f- x“) dx 
has likewise an algebraic sum. For it equals the former sum, with the addition of 
S J^x a d x, or a * Y (. v a + 1 + y a + 1 + z a + 1 -f- ), which is an algebraic quantity. 
In the same way we see that S J "* x-\-m x a -f- nx b + ••••) dx has an algebraic sum 
if m, n, a, b , &c. are constants, and if there are any number of such simple terms of 
the form m x°. Hence if the proposed integral be x . d x, and the supposition 
T 
p x or — = yp x does not succeed, we are led to try the suppositions 
— = x x, — = 4* x -f- x + T 2 , &c. &c. 
Example. Let the integral 4 x . d x = yv 1 -f- x n . d x as before. Suppose 
1 + x n = ~ -f- \f 1 + r, whence we deduce 
Q 
•*’” ~ x ~ s/ f + r . x — r =0, 
an equation which has n roots, whose product is r. We also find 
1 + r — 1 + 2 ' n — x *J 1 + x n + -j, 
+ i - v / t +/, 
whence 
x 
n — 1 
or 
p. v = x"-' +^-tx. 
And therefore since S j px dx has an algebraic sum = r -f- const., it follows that 
S J 4/ x dx has an algebraic sum also, viz. 
