Full text: Advanced calculus

42 
CALCULUS 
Hence 
/v»w 1 /y»m 
(2) d(x m y) = via dx + (m + l)b —dx+(m + l)c -— dx. 
V V y 
Let m + 1 = n, and integrate. Thus we obtain the reduction for 
mula : 
fx n dx _ x n ~ x y 2n — 1 b ¡'x n - l dx n — 1 a f*x n ~ 2 dx 
2n cj 
y 
y 
a f*x n ~ 2 dx 
C J y 
By taking d(y/(x — p) m ) and proceeding in a similar manner we 
obtain the reduction formulas 
(4) 
r dx 
J (x- p) n y ~ 
y 
2n — 3f'(p) i‘ dx 
(n - l)f(p) ( ic- p)»-i 2n - 2 f(p) J (x-p)”- 1 , 
— 2 c /' dx 
(x-p) n ~ 2 
_ n — 2 c 
n 
f(p) = « + + cp 2 =£ 0. 
(5) Ç dx — 2^ 2w - 2 c /* cfo 
t/(*- **)*? (« - i)/'G>)(® - p) n 2n - 1 f(p)J (x - py-'y’ 
f(p)= 0, f(p)*0. 
Reduction formulas for the integrals 
/g\ i* dx i* xdx 
J (x 2 + px + q) n y J (x 2 + px + q) n y ’ 
can be obtained by considering simultaneous! v 
p 1 — 4 q < 0, 
d [y(x 2 + px + g)-"] and d [xy(x 2 + px + g)-*]. 
EXERCISES 
1. Obtain a reduction formula for 
l dx 
r x n 
J Va 
+ bx 
2. Give the details of obtaining the reduction formulas (4) and (5). 
3. Obtain a reduction formula for 
/, 
dx 
(x 2 b 2 ) n Va 2 + x 1 
4. Obtain reduction formulas for the integrals (6). 
5. 
p 
V py 
6.
	        
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