. . t -.
: :
(∫f(x)dx)' = (∫f(x)dx)'x* x't = (∫f[φ(t)]* φ'(t)dx)' =
= f(x)* φ'(t) = f[φ(t)]* φ'(t) = f[φ(t)]* φ'(t)
. . . Þ . - .
- ., . - . . -. x=j(t) . t=y(x), y() - - .
. , .
.
(u v) = u v + u v, u = u(x); v = v(x).
d (u v) = (u v + u v)dx = v udx + u vdx =
d (u v) = v du + u dv.
. - ,
∫d (u v) = ∫v du + ∫u dv ∫d(u*v) = u*v
∫u dv = u*v - ∫v du
- , . - - . - dv - v, u dv , . . v du , . .
. : ∫xnexdx, ∫xn cosbx dx, ∫xn sinax dx, ∫xn lnx dx, ∫xn arcsinx dx, ∫xn arccosx dx .
. . . :
I. /(-), ,
II. /(-), k >1.
III. (+)/(2++q), (p2/u) - q = D < 0. . A, B, p, q - .
IV. (+)/(2++q), k > 1 - , (p2/u) - q = D < 0
. .
I. ∫ /(-) dx = A ln│x-a│+ C
II. ∫ /(-) dx = A ((x - a)-k+1/(-k+1)) + C
III. ∫(+)dx/(2++q) = A/2 ∫(2x+p)dx/(2++q) - Ap/2 - B ∫ dx/(2++q)
(2++q)' = 2x+p
Ax+B = (2x+p) A/2 - Ap/2 +B
. .
, , . - Þ -, .. ∫(2x+p)dx/(2++q) = ln│2++q│+ C
arctg, ∫dx/(2++q) = 2/√4q - p2 * arctg (2x+p)/√4q - p2 + C
:
∫(+)dx/(2++q)=A/2 ln│2++q│-(Ap-2B)/√4q - p2 * arctg (2x+p)/√4q - p2 + C
IV. . . -., k .
. . , . , .
. . . .
Q(x)/P(x) = M(x) + Q1(x)/P1(x)
. . . , . , . :
- . .
|
|
Q(x)/P(x) = P(x)/((x-a)α * (x2+px+q)β) =
. :
= A1/(x-a) + A2/(x-a)2 + Aα/(x-a)α + (M1x + N1)/(x2+px+q) + (M2x + N2)/(x2+px+q)2 + + (Mβx + Nβ)/(x2+px+q)β
. - . , . . 1 . . . 1 2 .
- (D<0), . .
. . 4 .