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.