.
- F(x) - f(x) D, D:F(x)=f(x).
F(x) - f(x) - D, - : ()=F(x)+c .
-. F(x) f(x), x D: F(x)=f(x).
() f(x), x . D: Ԓ(x)=f(x).
- φ()=()-F() - D → φ'()= Ԓ()-F()=f(x)-f(x)=0. - -, D → φ()=nst.=c → ()-F()==const → ()=F()+, ..
. - f(x) . . ∫f(x)dx=F(x)+c, f(x)- -,f(x)dx .
1)[f(x)dx]=f(x)
-: ∫f(x)dx=F(x)+c( .), (∫f(x)dx)= (F(x)+c)=F(x)=f(x)
2) d[f(x)dx] = f(x)dx
-: . : d[f(x)dx] = (f(x)dx)dx=f(x)dx
3)∫dF(x)=F(x) +c
-: ∫dF(x)=∫F(x)dx=∫f(x)dx=F(x)+c.
4) . :
∫kf(x)dx=k∫f(x)dx,x D, k R.
-: , k∫f(x)dx - k*f(x):
- : (k∫f(x)dx)=k*(∫f(x)dx)=k*f(x).
5)∫[f(x)+(-)g(x)]dx = ∫f(x)dx +(-)∫g(x)dx/
-: , ∫f(x)dx +(-)∫g(x)dx - [f(x)+(-)g(x)]:
- : [f(x)+(-)g(x)]= [∫f(x)dx] +(-)[∫g(x)dx]= f(x)+(-)g(x), ..
.
1) ∫ 0 dx = C = const 11) ∫dx/(√1-x2 )= arcsin x + C = - arccos x + C
2) ∫dx = x + C 12) ∫dx/(1+x2) = arctg x + C = - arcctg x + C
3) ∫xadx = xa+1/(a+1) + C, 13) ∫tgxdx = - ln |cosx| + C
a≠ -1 14) ∫ctgxdx = ln |sinx| + C
4) ∫dx/x = ln|x| + C 15) ∫ dx/(√a2- x2)=arcsinx/a +C=-arccos x/a + C
5) ∫exdx = ex + C 16) ∫dx/(a2+x2) = (1/a)arctg x/a + C=-(1/a)arcctg x/a + C
6) ∫axdx = ax/lnx + C 17) ∫dx/(x2a2) = (1/2a) ln |(x-a)/(x+a)| + C
7) ∫cosx dx = sinx + C 18) ∫dx/(a2-x2) = (1/2a) ln |(x+a)/(x-a)| + C
8) ∫sinxdx = - cosx + C 19) ∫dx/(√x2+A) = ln |x + (√x2+A)| + C
9) ∫dx/cos2x = tgx + C
20) ∫(√x2+A)dx = (x/2)(√x2+A) + (A/2) ln |x+(√x2+A)|+C
10) ∫dx/sin2x = - ctgx + C
21) ∫ (√a2- x2)dx = (a2/2) arcsin x/a + (x/2) (√a2- x2) + C
∫f(x)dx, x Î D
x = φ(t), t Î T, φ(t) T
, ∫ f(x)dx = ∫f(φ(t)) φ(t)dt, . . , ∫f(x)dx f(φ(t))φ(t)
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(∫f(x)dx)t( ) = (∫f(x)dx)x xt = f(x)φ(t) = f(φ(t))φ(t)
∫ f(x)dx = ∫f(φ(t)) φ(t)dt
u = u(x), v = v(x) D
d(uv) = duv + udv Þ ∫ d(uv) = ∫duv + ∫udv Þ uv = ∫vdu + ∫udv Þ
∫udv = uv - ∫vdu
:
1. Pn (x)φ(x)dx; Pn (x) n-
) φ(x) = sin ax u = Pn (x); dv = φ(x)dx
cos ax
eKx
b) φ(x) = u = φ(x); dv = Pn (x)dx
logax
2. ekxsin ax dx
ekxcos ax dx u
11. -:
y=f(x) [a;b] F(x)- f(x). :
∫(a b) f(x)dx= F(b)-F(a)
12. :
x=A(t) T X , f(x). F(x) f(x) X, F(A(t))- f(A9t))A(t) T, .. :
∫f(x)dx│x=A(t) = ∫f(A(t))A(t)dt
13. :
u(x) v(x) X. :
∫udv=uv-∫vdu
14. :
y=f(x) [a;b] (b>a). :
∫(a +∞)f(x)dx= lim b→∞ ∫( b)f(x)dx
, b .
15. :
, :
∫(b -∞)f(x)dx= lim a→-∞ ∫(b a)f(x)dx
16. :
17. Rⁿ
18. Rⁿ. .
Rⁿ, n>3, , Rⁿ . :
ρ (p,q)= │p-q│=√(x1- x1)²++(xⁿ-xⁿ)²
, p=(x1, x2, , xⁿ) q=(x1, x2, , xⁿ) Rⁿ.
:
1) ρ (p,q)>0, p ≠ q, ρ (p,p)=0;
2) ρ (p,q)= ρ (q,p);
3) ρ (p,q)+ ρ (q,r)>= ρ (p,r), p,q r. ( ).
19. Rⁿ.
pₒ- Rⁿ ε . , ε pₒ , pₒ ε:
{p Rⁿ │ ρ (pₒ,p)< ε}.
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ε pₒ B(pₒ, ε) U3(pₒ). U3(pₒ)
ε pₒ.
20. :
Rⁿ. :
- ,
ε;
- , Rⁿ;
- , , , , , , .
21. .
X , .
X , .
22. .
X - Rn. p0 X,
ε- p0 X, p0.
p0 , X.
p0 Î X X,
ε-, X, p0, .
, ,
.
23. .
X Rn , .
, , C>0, p=(x1,x2,,xn) : |x1| .
24. Rn, .
Rn. , p0, 0.
p1=(x1,y1), p2=(x2,y2),- . , p0=(x0,y0), x1,x2, x0, y1,y2, - y0.
25. .
?
26. () .
f(x,y)=C , f C.
n>2 , . f(
27. .
X Rn f(p) p0 . a f p0, p0 , pn p0, .
: , :
28. .
f(p), X Rn,
0 Î X, , , p0 .
f(p), X Rn, , .
29. , : , .
f n Rn, .
f n Rn, p0 Î X, f , q0 Î X, f .
30. .
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, .
z=f(x,y) (x0,y0) :
zx, dz/dx, fx(x0,y0) x;
zy, dz/dy, fy(x0,y0) y.
31. D Rn Rn, (x1, x2, ., xn) (tx1, tx2, ., txn) t>0 f(x1, x2, ., xn) D λ, t>0 f (tx1, tx2, ., txn)=tλ f(x1, x2, ., xn).
, . 2 . =t2
32. 3:
F (x,y)=x2
F (tx, ty)=t2x2√(tx*ty)=t3 F (x,y)
33. f (tx1, tx2,tx3)=tλ f(x1, x2, x3). u= f(x,y,z)
34. z=f(x;y) D (0;0) D ( D), (), e - , , (0;0) :
f(x;y)>f(0;0) min
f(x;y)<f(0;0) max