- ( )

  1. . '
  2. " Ѳ ² ղ ²ʲ
  3. .
  4. ' ֲ ˲

P (x) / Q (x) - / . R, degP (x) .

=> P = Q * q + r, P / Q = q + r / Q, r / Q -

: - ( ? 0).

: P1/ Q1+ P2/ Q2= P1Q2+ P2Q1/ Q1Q2

degP11? degP1Q21Q2

degP22? degP2Q12Q1

deg (P1Q2+ P2Q1) ? deg [max (P1Q2, P2Q1)] 2Q1

: 1 / (x-l)k, 1 / x2+ Px + q, Ax + B / x2+ Px + q

1)

a2= Q - p2/ 4> 0 => (x2+ Px + q) <0 b = x + p / 2 c = b / a

2)

3)

P (x) / Q (x) - (Q (x) - )

z1, z2, z3, ..., ZnI

Th: P (x) & Q (x) -

degP (x) 1) (X-z2) (X-z3) ... (X-zn) zi? zj i ? j

$! A1... ANI C:

: A1... AN :

P (x) = ,

P (xi) = Ai* Qi(xi) AI = P (xi) / Qi(xi) -

A1... AN, Ai= P (xi) / Qi(xi) , .

:

A1... AN, Aj = P (xi) / Qj(xi) => P (x) - = R (x), R (x) ? 0 degR (x) ? n-1

R (zj) = P (zj) - = P (zj) - AJQJ(zj) ( AIQI(zj) = 0 i ? j)

R (zj) - P (zj) + AJQJ(zj) = 0. degR (x) J(X) degR (x) - P (x) + AJQJ(X) JQJ(X) n (z1, ..., Zn- ) => .

: P Q I R [x], Q (x) = (x-z1) (X-z2) (X-z3) ... (X-zs) (X2+ xp1+ q1) (X2+ xp2+ q2) ... (X2+ xpk+ qk), s - , k - , Q , $ A1... AS, B1... BK, C1... CKI R :

P / Q = A1 / X-z1+ ... + AS / X-zs+ B1x + C1 / (X2+ xp1+ q1) + ... + BKx + CK / (X2+ xpk+ qk)

wi wi '- ' x2+ xpi+ qi=>

Q (x) = (x-z1) (X-z2) (X-z3) ... (X-zs) (X-w1) (X-w1 ') ... (X-wk) (X-wk ') => :

P / Q = A1/ X-z1+ ... + AS/ X-zs+ E1/ X-w1+ F1/ X-w1 ' + ... + EK/ X-wk+ FK/ X-wk '

' :

P '= P, Q' = Q, zi '= zi

P / Q = A1 '/ X-z1+ ... + AS '/ X-zs+ E1 '/ X-w1 '+ F1 '/ X-w1+ ... + EK '/ X-wk+ FK '/ X-wk

( ) :

AI '= AI, I = 1, ..., s => AII R

EJ= F 'J, J = 1, ..., k

EJ/ X-wj+ FJ/ X-w 'j= (EJ+ FJ) X - (FJ* wj+ EJ* W 'j) / (X2+ xpj+ qj) =

(F 'J+ FJ) X- (FJ* wj+ F 'J* W 'j) / (X2+ xpj+ qj) = (F 'J+ Fj) X - ((FJ* wj) + (FJ* wj) ') / (X2+ Xpj + qj)

(F 'J+ FJ) = B I R

(FJ* wj) + (FJ* wj) '= C I R

9. oR (sinx, cosx) dx

ϳ: u = tg x / 2-

x = o2arctg u, dx = 2du / 1 + u2

R (sinx, cosx) dx = oR (2u / 1 + u2, 1-u2/ 1 + u2) 2du / 1 + u2- - .

8. oxm(A + bxn)pdx

: x = t1 / n => Dx = 1 / n * t1 / n - 1dt oxm(A + bxn)pdx = 1 / n * o (a + bt)p* t(M + 1 / n - 1)dt

.., oxm(A + bxn)pdx x = t1 / n : o (a + bt)p* tqdt, p q , q = m + 1 / n-1

1 )p - .

q = r / s, r, s - , z = t1 / s(T = zs) : dt = sz(S-1)dz => o (a + bt)p* zr* sz(S-1)dz = so (a + bt)p* z(R + s-1)dz

2 )q -

p = r / s, r, s - , z = (a + bt)1 / s : t = (zs- A) / b => dt = (1 / b * szs-1) Dz => ozr* ((Zs- A) / b)q* (1 / b * szs-1) Dz = o (zs- A) / b)q* S / b * z(R + s-1)

3 )p + q -

p = r / s r s . o (a + bt)p* tqdt = o ((a + bt) / t)p* t(Q + p)dt, z = ((a + bt) / t)1 / s . : t = a / (zs-b) =>

dt = -asz(S-1)/ (Zs-b)2 => O ((a + bt) / t)p* t(Q + p)* -asz(S-1)/ (Zs-b)2 = -a(Q + p + 1)soz(R + s-1)/ (Zs- B)q + p + 2




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