Full text: Advanced calculus

202 
CALCULUS 
can be referred to an integral already treated in § 2 by means of 
the transformation 
x — c = z 2 or = — z 2 , 0 ^ z, 
according as the values of x between x 0 and x 4 make x — c positive 
or negative. 
Example. Consider the integral: 
i 
j V— (x 
dx 
(x — l)(x — 2)(x — 3) 
Let x — 1 = — z 2 . The integral thus goes over into 
o 1 
' T —2 dz o r dz 
J V(1 + z 2 )(2 + z 2 ) J V(1 + z 2 )(2 + z 2 ) 
The substitutions x — 2 = — z 2 and x — 3 = — z 2 would have led 
to other forms equally tractable. 
dx 
4. The General Case, 
j- 
V Gi(x) 
Let (r 4 (a,-) be a polynomial of the 4tli degree, whose roots or factors 
are all distinct. If Cr 4 (x) has a real root, x = a, the transformation 
(1) 
y = 
x = a + -, 
y 
will carry the integral into an integral of the form treated in § 3, 
namely: 
cly 
/: 
V Gs(y) 
It remains, therefore, merely to discuss the case that 
(2) G\ (x) = (x 2 + p t x + <p) (x 1 + p 2 x + q 2 ), 
0 < 4 g 4 — pf, 0<4 q,-pl, 
the second factor not being identical with the first. Let 
(3) x ;= y + h. 
Then x 2 +p l x + q l = y 2 +p[y + q[, x 2 +p 2 x + q 2 =y 2 +p! 2 y + q! 2 , 
where q[ = h 2 + pji + q u q'i = № + + Qi • 
Let us seek to determine h so that q[ and q 2 will be equal:
	        
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