.
:
Pn(x) = xn + an1xn+1 + an-2xn+2 +... + a0 = (x - a1) (x - a2)... (x - an)
{ . ax2 + bx + c = a(x x1)(x x2) }
n n
(x - ai), ai - . , , .
a = a + ib, a* = a - ib
(x - a)(x - a*) = (x a ib)(x a + ib) = (x a)2 + b2 = x2 + px + q,
D < 0, .
Pn(x) = (x - a1)k1(x - a2)k2... (x2 + p1x + q1)r1..., k1+k2+...+2(r1+r2...) = n.
. , n < m. n > m,
= , k < m
: 1) -
2) = (x + 6) +
: , , , , D < 0
:
1) J1 = ò dx = A ò = A ln (x a) + C
2) J2 = ò dx = B ò = B ò = B + C
3) J3 = ò dx, D < 0. :
1. x2+px+q = (x+p/2)2 + (qp2/4);
2. t = x + p/2; 3.
Ja = = = , Jb = =
4) J4 = ò dx.
J `a = = , J `b = . Jb , =
. J = ; D = -36 < 0,
: 1) x2 + 2x + 10 = x2 + 2x + 1 + 9 = (x + 1)2 + 32 ;
2) t = x + 1, x = t 1, dx = dt, x2 + 2x + 10 = t2 + 32,
3) J = = + 2 = 3/2 ln|t2 + 32| + 2/3 arctg t/3 + C
= 3/2 ln| x2 + 2x + 10| + 2/3 arctg (x 1)/3 + C
. J = = =
= = + 2 = 3/2 ln|t2 + 32| + 2/3 arctg t/3 + C
.
) - .
Rm(x) = (x - a1)(x - a2)2(x - a3)3.. m .
. - :
+ . - : + +
= + + + + + +...
, Rm(x), Tn(x). , Bi, Ci Tn(x) = Pn(x). . .
.
1 . Tn(x) Pn(x) , n , , Bi, Ci
.2.
x1 | A + B = 2 Þ A = -5
x0 | -2A B = 3 Þ B = 7
2 . Tn(x) = Pn(x) ai .
.2.
= 1, 5 = - A Þ A = -5
|
|
= 2, 7 = B Þ B = 7
) - .
x2+px+q, D < 0, . . - : + .
.3.
x2 | A + M = 0; x1 | N M = 2; x0 | 4A - N = 1 Þ A = 3/5; M = -3/5; N = 7/5.
.
sinmx cosnx dx, m n (8)
. sin3x dx = sin2x sin x dx = - (1-cos2x)d(cos x) = - (1t2)dt = -cos x + cos3x/3 +C
sin2x cos x dx,
sinmx cosnx dx, m n (9)
sin2x = ½ (1 cos 2x); cos2x = ½ (1 + cos 2x); sin x cos x = ½ sin 2x
tg x = t (. )
. sin2x dx = ½ (1 cos 2x) dx = ½ dx - ¼ cos 2x d(2x) = x/2 - ¼ sin 2x +C
sin2x cos2x dx, cos4x dx
sin ax cos bx dx (10)
sin a cos b = ½[sin(a+b) + sin(a-b)]; cos a cos b = ½[cos(a+b) + cos(a-b)]
sin a sin b = ½[cos(a-b) cos(a+b)]
. sin 3x cos x dx = ½ (sin 4x + sin 2x)dx = -1/8 cos 4x - ¼ cos 2x + C
sin 5x cos 3x dx
R(tg(x), sin2n x, cos2mx) dx tg x = t, (11)
= (1 + tg2x) dx = dt dx = dt/(1+t2), x = arctg t
cos2x = = , sin2x = (1 cos2x) =
. tg3x dx = t3/ (1+t2) dt = (t3 + t t)/ (1+t2) dt = t dt - t/(1+t2) dt =
= t2/2 - ½ d(1+t2)/ (1+t2) = ½ tg2x ½ ln |1+tg2x| + C
. = = (t -4 + t 2) dt = - (tg x)-3 /3 - (tg x) 1 + C
tg x/2 = t R(sin x,cos x) dx , sin x, cos x R() 1- , sin x = , cos x = , dx =
.
{ sin x = 2sin x/2 cos x/2 = 2 cos2x/2 = ; (12)
cos x = cos2x/2 sin2x/2 = cos2x/2 (1 - ) = cos2x/2 (1 t2) = }
. = { 1 + sin x = 1 + = } = 2 =
C
;
.
. . , , .
. ,
R( x, , ,... ) dx (13)
(k1, k2,..., km) = n, n/ki - x = tn .
. J = = { (2,3) = 6, x = t6, dx = 6t5dt, t = x1/6 } =
= = 6 = 6 =
= 6(t3/3 - t2/2 + t) 6 ln |t + 1| + C = 2 x1/2 - 3 x1/3 + 6 x1/6 - 6 ln | 1 + x1/6 | + C
R( x, , ,... ) dx (14)
(k1, k2,..) = n, ax + b = tn, dx = (n/a)tn 1dt, x =(tn b)/a .
. J = = { (1,2) = 2, 3 x = t2, dx = - 2 tdt, x = 3 t2 }= = - 2 = -6 t3/3 + 2 t5/5 + C = -2 (3 x)1/2 + 2/5 (3 x)5/2 + C
;
- R( x, , ,... ) dx (15)
(k1, k2,..) = n, = tn, .
. J = = { = t2, x = , dx = } =
= 2 ; Þ A = B = D = ¼, C = - ¼;
J = ¼ { ln| t 1 | - ln| t+1 | - (t 1)-1 (t + 1)-1 } + C
R( xm, ) xm -1 dx (16)
|
|
xm 1dx xm d(xm) d(axm + b), axm+ b = tn .
xm-1 dx = n tn-1 dt/a, xm = (tn-b)/a.
. J = = { 2x2+1 = t3, xdx = ¾ t2 dt, x2 = ½(t3-1) } =
= ¾ ½ (t3-1)t2 dt/t = 3/40 t5 3/16 t2 + C
. . (17)
) R( x, ) dx; ) R( x, ) dx; ) R( x, ) dx
.. 2 .
) : x = a sin t Þ a2 x2 = a2 (1 sin2 t) = a2 cos2t, = a cos t
t = arcsin (x/a), dx = a cos t dt
) : x = a tg t Þ a2 + x2 = a2 (1 + tg2 t) = a2 / cos2t, = a / cos t
t = arctg (x/a), dx = a dt / cos2 t
) : x = a / cos t Þ x2 a2 = a2 (1 / cos2 t 1) = a2 tg2t, = a tg t
t = arcos (a/x), dx = - a sin t dt / cos2 t
. J = = { x = 2 sin t, dx = 2 cos t dt, = 2 cos t }=
= - ctg t - t + C = - ctg (arcsin(x/2)) - arcsin(x/2) + C
. J = = { x = 2 tg t, dx = 2 dt / cos2 t, = 2/cos t }=
= 8 - 8 8/3 cos-3 t + C = 8/3 cos-3(arctg(x/2))+ C
. J = = { , , = 2 tg t } =
= - = - ½ = - ¼ = - ¼ (t - ) + C =
= - ¼ [ arcos (2/x) + ½ sin (2(arcos(2/x))) ] + C
. . (18)
R( x, ) dx :
1) : ax2 + bx + c = a[ x2 + px + q ] =
= a[ x2 + 2x(p/2) + (p/2)2 - (p/2)2 + q] = a [ (x + p/2)2 + (q p2/4) ], p = b/a, q = c/a. 2) : x + p/2 = z, ax2 + bx + c = |a|{ z2 + (q p2/4)}
: ), ), )
. J = = { x2 + 2x 3 = (x +1)2 4 = t2- 22, t = x +1, dx = dt }
= = { (): , , } =
= -4 = -4 ; +
Þ A = B = D = ¼, C = - ¼; J = - { ln + 2 }+ C, z = arccos .