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.