.
1. .
2. .
3. , .
4. . .
5. .
6. . .
7. , . .
8. , ( ).
9. .
, 8 . , .
1. .
1. ( ).
2. =0, =0. =0, . , (0;0) .
3. (-) = , .
4. . .
5.
k b, . . , .
6. : =1 . , y' = 0 . .
, (- (1;+ , (0;1) .
max = f (0)= 0, ymin = f (1) = -1.
7.
'' '' =0. =0 . .
, ( , .
8. , , (0;0) .
20. .
.
f(x), f(x) - , (), dx - x, dx.
. F(x) + C, C, f(x)dx, .. . .
, - :
, . , .. .
, , , , , - .
|
|
f (x) X, k ,
: .
f (x) g (x) X,
: .
f (x) X, :
: .
f (x) X , :
:
.
22. .
|
v (x) V (x).
∫ U (x) v (x) dx = ∫ U (x) V '(x) dx = ∫ U (x) dV (x). |
, v (x) V (x).
U (x) V (x) U (x) = w (V (x)),
∫ U (x) dV (x) = ∫ w (V (x)) dV (x) = ∫ w (t) dt, |
t = V (x).
∫ w (t) dt
t = V (x) , .. .
U (x) V (x) U (x) = w (V (x)), , . .
1. U (x) V (x) ∫ V (x) U '(x) dx.
∫ U (x) V '(x) dx
∫ U (x) V '(x) dx = U (x) V (x) − ∫ U '(x) V (x) dx. | (1) |
. .. .. : . . .: , 2000. . 168.
1. , , , V U. .
2.
∫ U (x) v (x) dx = U (x) V (x) − ∫ u (x) V (x) dx, | (2) |
v (x) V (x), U (x) , u (x) = U '(x) , .
3. (1)
∫ U (x) dV (x) = U (x) V (x) − ∫ V (x) dU (x). | (3) |
:
1. ln x, arcsin x, arccos x, arctg x. (2), U (x) , Vxu (x) .
|
|
2. : Pn (x) eαx, Pn (x)sin αx P (x)cos αx, Pn (x) n.
n (1), U (x) . .
3.
eαx cos βx, eαx sin βx, sin(ln x), cos(ln x). |
.
4. ∫ U (x) dV (x), U (x) V (x), V (x) U (x).