2.1. topshiriq

Quyidagi
yeching:
 x1 + 2 x2  2

1.  x1 + x2  6
2 x + x  11
2
 1
I topshiriq
Chiziqsiz programmalashtirish masalasi
chiziqsiz programmalashtirish masalalarini grafik usul bilan
x1  0, x2  0
Z = 2( x1 − 7)2 + 4( x2 − 3)2 → min (max)
x + x  7
3.  1 2
2 x1 − x2  8
x1  0, x2  0
Z = 4( x1 − 2)2 + 2( x2 − 2)2 → min (max)
 x1 + 2 x2  8

5. 3 x1 + x2  15
x + x  1
2
 1
x1  0, x2  0
Z = ( x1 − 6)2 + ( x2 − 2)2 → min (max)
3x1 + 2 x2  12

7.  x1 − x2  6
x  4
 2
x1  0, x2  0
Z = 9( x1 − 5)2 + 4( x2 − 6)2 → min
 x1 + 2 x2  2

2.  x1 + x2  6
2 x + x  10
 1 2
x1  0, x2  0
Z = x1  x2 → max
 x1 + x2  2
 x − x  −2
 1 2
4. 
 x1 + x2  6
 x1 − 3x2  2
x1  0, x2  0
Z = 25( x1 − 2)2 + ( x2 − 2)2 → max
6 x1 + 4 x2  12

6. 2 x1 + 3x2  24
−3x + 4 x  12
2
 1
x1  0, x2  0
Z = x1  x2 → max
3 x1 + 2 x2  7

8. 2 x1 − x2  8
−3x + 4 x  12
2
 1
x1  0, x2  0
Z = ( x1 − 3)2 + ( x2 − 4)2 → min (max)
x  x  8
9.  1 2
0  x1  6
x + x  1
10.  1 2
2 x1 + 3x2  12
0  x2  4
x1  0, x2  0
Z = x1 + 3x2 → min (max)
Z = ( x1 − 4)2 + ( x2 − 6)2 → min
11. x12 + x22  16
12. x12 + x22  16
x1  0, x2  0
x1  0, x2  0
Z = 2 x1 + x2 → min (max)
2 x1 + 3 x2  6

13. 3 x1 − 2 x2  18
− x + 2 x  8
2
 1
Z = ( x1 − 2)2 + ( x2 − 1)2 → min (max)
 x1  x2  2
14. 
 2
2

 x1 + x2  16
x1  0, x2  0
x1  0, x2  0
Z = ( x1 − 4)2 + ( x2 − 3)2 → min (max)
Z = 3x1 + x2 → min (max)
 x1 − x2  4

15. 2 x1 + x2  8
x  6
 2
5 x + 4 x2  −20
16.  1
3x1 + 2 x2  18
x1  0
x1  0, x2  0
Z = ( x1 − 4)2 + ( x2 − 2)2 → min (max)
Z = ( x1 − 5)2 + ( x2 − 4)2 → min (max)
Quyidagi chiziqsiz programmalashtirish masalalarini Logranj ko’paytuvchilar
usuli bilan yeching:
17. z =
1 1
+
x y
18. z = x − y
19. z = xy 2
x + y = 2.
x 2 + y 2 = 1.
x + 2 y = 4.
20. z =
x− y−4
2
21. z =
1 1
+
x y
x 2 + y 2 = 1.
x + y = 2.
22. z = x 2 + 6 x − 2 y + 1
23. z = 2 x 2 + y 2
z = e xy
24.
25. z =
x + y − 2 = 0.
x + y = 1.
x + y = 2a ( a  0 ).
1 1
+
x y
x 2 + y 2 = 1.
26. z = xy
x 2 + y 2 = 1.
z = 6 − 4x − 3 y
27.
28. z =
1 1
+
x y
1
1
+ 2 = 1.
2
x
y
29. z = x 2 + y 2
x + y = 1.
x + y = a.
z = e xy
30.
x 2 + y − 4 = 0.
II topshiriq
Aniqmas integrallarni hisoblang:
1.
x cos ( x ) dx ;
(
e 2 x−1
dx.
2x − 1

2
)
2
x
a
−
xdx
,
a
−
x
=
t

3
2. x3 1 − 2 x 4 dx ;
3.
dx
 x ln x ;
4.


sin 2 x
2 + cos x
2
cosx dx
1 + 2sin2 x
dx ; ∫(𝑥 + 1)𝑒 𝑥 𝑑𝑥
5. ∫ 𝑒 2𝑥 cos 𝑥 𝑑𝑥 ; ∫
6. ∫ 𝑥 2 𝑒 3𝑥 𝑑𝑥 ;
7. ∫ 𝑙𝑛2 𝑥𝑑𝑥;
.
ln 𝑥
𝑥3
𝑑𝑥
∫ 𝑥 𝑐𝑜𝑠𝑥𝑑𝑥
∫
𝑥 cos 𝑥 𝑑𝑥
𝑠𝑖𝑛3 𝑥
8. ∫
9. ∫
𝑥
2
√22 −𝑥 2
𝑎𝑟𝑐𝑠𝑖𝑛
𝑑𝑥
𝑑𝑥; ∫ 3
𝑥
𝑑𝑥
𝑑𝑥
; ∫ 2
2𝑥 −6
√2−𝑥 2
2𝑥+3
10. ∫(1 + 𝑒 𝑥 )2 𝑑𝑥 ;
∫ 𝑥 2−5 𝑑𝑥 ;
2𝑥 𝑑𝑥
11. ∫ 𝑡𝑔2 𝜑𝑑𝜑;
∫ 𝑥 4+3
sin 𝑥 𝑑𝑥
𝑑𝑥
; ∫ 𝑥 ; ( 𝑒 𝑥 + 1 = 𝑡 2 );
√𝑒 +1
√1+2 cos 𝑥
12. ∫
𝑒 𝑥 𝑑𝑥
13. ∫
3+4𝑒 𝑥
; ∫ 𝑒 sin 𝑥 cos 𝑥 𝑑𝑥;
√2−𝑥 2 +√2+𝑥 2
14. ∫
√4−𝑥 4
𝑑𝑥; ∫ 𝑥 𝑠𝑖𝑛𝑥 𝑑𝑥;
15. ∫ 𝑥 3 𝑒 𝑥 𝑑𝑥; ∫ √𝑎2 − 𝑥 2 𝑑𝑥, 𝑎 > 0, 𝑢 = √𝑎2 − 𝑥 2 ;
∫(𝑥 2 + 2𝑥 + 3) cos 𝑥 𝑑𝑥 ;
16. ∫ 𝑠𝑖𝑛√𝑥𝑑𝑥; √𝑥 = 𝑡;
3−2𝑐𝑡𝑔2 𝑥
17. ∫
18.

19.
 3x +1 ;
20.
𝑑𝑥.
𝑐𝑜𝑠 2 𝑥
5
xdx

x2 + 2
dx
2 x −1
;
dx
 (4 x + 3)
3x + 2dx ;

dx
e
5
dx
2− x
x 2 dx
 2x3 + 5
;
21.  9 x + 1 / 4) sin(2 x 2 + x)dx ;
22.  e −
2x
dx
;
23.  cos 2 x + 1 dx ;
5
24.
25.


3
arctgx / 3
9+ x
2
dx
3 + 2x 2
dx ;
;
x2 +1
dx ;
26. 
x +1
e
x

2 + 5e x dx
sin ln x
dx
x
dx
2
+1
 2x
e−x
 1 − e −2 x dx
x 3 dx
 1− x

3
2 + cos 3x sin xdx
27.
2x − 1
xdx
28.

29.
 3x
30.
dx
2
−5
 2 x + 1 dx ;  3x
16 − x 4
2x + 1
dx ;
2
+2

x 2 dx

;

x + ln 2 x
dx ;
x
x6 +1
2x + 1
x2 +1
dx
xdx

x2 + 2
III topshiriq
Boʻlaklab integrallash usulidan foydalanib integrallarni hisoblang:
1.  xe 5 x dx
2.  x 2 e − x / 2 dx
3.  x 3 e 2 x dx.
4.  ln(1 − x)dx
5.  ( x 2 − 3x) ln xdx
6.  x 2 ln 2 xdx
8.  x sin 3xdx
9.
11. 
2 − x 2 dx
12.  x cos 2 xdx

arcsin x
7.

10.
ln(1 − x)
x

dx
x 2 − 4dx
x
 cos
2
x
dx
13.  arctg 7 x − 1dx
14.
16.  e x sin x dx
2
17.  cos(ln x)dx
18.  e x dx
19.  ln( 1 − x + 1 + x )dx
20.  x cos2 x dx.
21.  xtg 2 2 xdx
22.  x ln 1 − x dx.
23.  cos 2 (ln x)dx
24.  x 2 arctg 3xdx.
x
25.  arcsin
dx
2
x
26.  (arcsin) 2 dx.
27.  3 x cos xdx.
28.  e 3 x sin 2 xdx.
29.  ln(1 + x 2 )dx.
30.
x
dx
sin x
1+ x
IV topshiriq.
Integralni hisoblang.
1.  2 dx
x −x−2
1
 (2 − x)
2

2− x
dx.
2+ x
15.  x 2 cos dx
ln x
 ( x + 1)
2
dx
2.
x2
 (1 − x) 3 dx
3.  3 dx 2
x −x

( x − 1) ( x − 2 )
3
3x − 5

4.  3dx
x +x
dx

dx
( x + 1)
+ ( x + 1)
5.  3dx
x +1
 sin x cos
6.  xdx
x3 −1
 sin
dx
7.
 x(x + 1)
8.
 (x
9.
( x 2 + 2)dx
 ( x + 1) 2 ( x − 1)
2
3
2
1
2
.
2 xdx.
xdx
dx
 sin x − cos x
2
( x − 2 ) dx

dx
− 1)( x + 2)
x − 10 x + 29
2
dx


10.  2 xdx
x + 3x − 4
3
1 − 2 x − x2
 (
x 1+
x3 + 1
dx
12.  3
x − 5x 2 + 6 x

(
) dx.
3
x
x
dx
3
x 4 + 3x 3 + 2 x 2 + x + 1
dx
14. 
x2 + x +1

.
1+ x +1
dx.
1+ x −1

3x 2 + 8
dx
13.  3
x + 4x 2 + 4x
15.  4 dx 2
x +x
.
x3 − 4  x 2dx.
x2 − x
dx
11.  2
x − 6 x + 10
16.  4dx
x +1
dx.
x2 − 4 x + 5
3
2
.
 3 2 − x3
5
3
 x  3 (1 + x ) dx.
2
x −1
dx.
2x − 1
dx
x 1− x
)
.
2
.
17.
 (x
3x + 5
dx
+ 2 x + 5) 2
2
dx
 (4 + x
20.
x3 − 3
 x 4 + 10 x 2 + 25 dx

2 2
)

23.

24.

3

dx
3
( x − 1) ( x + 1)
x2 − x

.
dx
.
x+4x
dx
(
x  1+ x
3
( 2 x + 1)
− 2x + 1
dx
 1 + 3 x + 1.
xdx
1− 3 x .
27.
xdx
 x 2  x − 1.
xdx
1+ x .
28.
 (
29.
30.
x
3
x
dx
x +1
4
2
dx
x+
3
x2
.
.
)
xdx
x−
4
dx.

x dx
dx.
x +1 + 3 x +1

.
25.
x 1+
.
x+2
dx.
x

2
)
3
.
.
1 − x2
dx
3
2
dx
xdx
.
 1 + x 2 dx.
3
 (1 − x )
21.  4dx
x −1
22.
 x4 + 1
dx
.
1 − 2x − 4 1 − 2x
19.
3
11
x
18.  4 dx2
x + x +1
dx
x
3
x
2
.
x + 3 x2 + 6 x
1+
(
x 1− 3 x
x
4
x3
dx.
)
dx.
V topshiriq.
Aniq integralni hisoblang.
e2
1.
2 ln x + 1
e x dx
ln 8
2.
e
3
8.  arctgx dx
1
3.  x 2 1− x 3 dx
−2
e
6.  x ln x dx
1
1
9.  x 2 e −2 dx
−1
0
 2

10.  e x sin x dx
11.
0
 cos
7
2
14.
x sin x dx
0
ln 2
e − 1 dx
17.
x
0
3
x
1
0

−1
x 4 sin x
 x 6 + 2 dx
−7
dx
0 x 2 + 4x + 5
12
dx
20.
x + 5x + 1
2

0
1+ x
dx
1− x
 4
dx
3
1+ x +1
23.
3
 tg x dx
 2
15.  sin 3 x dx
0

18.
2
21.
П
6
−3

0
dx
29. 
x )2
1 (1 +
dx
+ 2x
3
27.
4
dx
28.  2
П cos 2 x
2
24.  x 2 9 − x 2 dx
3
dx
26.  2
x
2
0
x
1
3
25.  x 2 cos xdx
dx
 3 + 2 cos x
0
0
 /2
 x sin x cos x dx
−
1

16.
 (x + 3)sin x dx

12.
0
 2
13.
22.
x
 x e dx
1
0
7.  ln 2 x dx
19.
dx
2x + 1
ln 2
5.
ex +1
ln 3

0
e x dx

4.
4
xdx
4 − x2
1
e x dx
30. 
1+ e2x
0
8
VI topshiriq
Berilgan chiziqlar bilan chegaralangan figuralar yuzalarini hisoblang:
1. y = 4 x − x 2 parabola va Ox o‘qi bilan chegaralangan figura yuzi aniqlansin.
y2
= 1 giperbola bilan chegaralangan figura
2. y = ( x − 1) parabola va x −
2
yuzi hisoblansin.
3. Tenglamasi x = 2(t − sin t ), y = 2(1 − cos t ) bo‘lgan sikloidadaning bir arki va
2
2
Ox o‘qi bilan chegaralanga figura yuzi aniqlansin.
4. Tenglamasi x = a cos3 t , y = a sin 3 t bo‘lgan asroida bilan chegaralangan figura
yuzi aniqlansin
5. y = e x , y = e x/2 , y = e2 .
6. y = x 4 − 2 x 2 , y = 0.
7. y = 3 + 2 x − x 2 , y = x + 1.
8. y = x2 + 3, xy = 4, y = 2, x = 0.
9. y = 1 − x , y = x + 1.
10. y = cos2 x, y = 0, x = 0, x =  / 4.
11. x = 0, x = 2, y = 2 x , y = 2 x − x 2 .
12. y = arcsin 2 x, x = 0, y = − / 2.
13. y = x2 + 1, x = y 2 , 3x + 2 y − 16 = 0, x = 0.
Egri chiziqlar yoylari uzunliklari hisoblansin:
15. y = 2 x x = 0 dan x = 1 gacha.
16. y = ln x x = 3 dan x = 8 gacha.
17. x = t − sin t , y = 1 − cos t t = 0 dan t = 2 gacha.
Ox oʻqi atrofida aylantirishdan hosil boʻlgan aylanma jism sirtining yuzini toping:
3
19. y = x , x  0 ; 4 1/ 3  .
20. 9 y 2 = x ( 3 − x )2 , x  0 ;3.
21. x 2 + y 2 = 9, x   −2 ;1.
1
22. y = x3 (0  x  )
2
 
23 . y = cos x 0; 
 2
24. x = a(t − sin t ), y = a(1 − cos t ) sikloidaning bir arkini Ox
o‘qi atrofida
aylantirishdan hosil bo‘lgan sirt yuzasi topilsin.
25. x = a cos3 t , y = a sin 3 t astroidaning Ox o‘qi atrofida aylantirishdan hosil
bo‘lgan sirt yuzasi topilsin.
4
giperbola, x = 3 va x = 12 to‘g‘ri chiziqlar hamda absissalar o‘qi bilan
x
chegaralangan chiziqli trapetsiyani Ox o‘qi atrofida aylantirishdan hosil bo‘lgan
aylanma jism hajmi aniqlansin.
x
27. y = , x = 4, x = 6 va absissalar o‘qi bilan chegaralangan trapetsiyani Ox o‘qi
2
atrofida aylantirishdan hosil bo‘lgan aylanma jism hajmini toping.
28. 4 x 2 + 9 y 2 = 36 ellipsni kichik o‘qi atrofida aylantirishdan hosil bo‘lgan aylanma
26. y =
jism hajmi topilsin.
29. y = log 2 x, y = log 4 x egri chiziqlar va y = 1 to‘g‘ri chiziq bilan chegaralangan
sohani Oy o‘qi atrofida aylantirishdan hosil bo‘lgan aylanma jism hajmini toping.
30. 4 x 2 + 9 y 2 = 36 ellipsni kichik o‘qi atrofida aylantirishdan hosil bo‘lgan aylanma
jism hajmi topilsin.