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B 4 4 4 52 4/ 9/ 3/3 6 9.. y = x 2 x x =

(, ) (, ) S = 2 = 2 4 4 [, ] 4 4 4 ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, 4 4 4 4 4 k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) 2 2 + ( ) 3 2 + ( 4 4 4 4 4 4 4 4 4 ( ( = ) 2 ( ) 2 2 ( ) 3 2 ( ) ) 4 2 + + + 4 4 4 4 4 = 4 ( ) k 2 4 4 = 4 k= 6 = 4 6 4 k= 4 6 k 2 = 5 32 =.46 (4 + )(2 4 + ) n ( 2 + 2 2 + + n 2 ) = k 2 = n (n + )(2n + ) 6 k= ) 2 2

8 4 S S 4 8 2. [, ] 8 S 8. 2. 3. S 8 n n 3. [, ] n S n. 2. 3. S n 3

n n S n lim S n n lim S n = lim n n = lim n (n + )(2n + ) 6n2 ( + ) ( 2 + 6 n n = 6 2 = 3 ) lim n S n 3 [, ] 3 y = x2 x x = 3 S n lim S n n 3 [, ] 4 4 4 ( ) k 2 k =, 2, 3 4 s 4 s 4 = 4 3 k= ( ) k 2 = 4 4 6 3 k= k 2 = 7 32 4

4. [, ] 8 s 8. 2. 3. s 8 5. [, ] n s n. 2. 3. s n n s n lim n s n lim s n = lim n n = lim n (n )(2n ) 6n2 ( ) ( 2 6 n n = 6 2 = 3 [, ] 3 n s n < S < S n lim s n = lim n S n = 3 y = x 2 x x = S 3 S x 2 dx ) 2 y = x 2 f(x) y = f(x) x x = x = b < b 5

f(x) =x x x 2 x 3... x n =b [, b] n = x, x, x 2,..., x n, x n = b x k x k (k =, 2,..., n) f(x k ) (k =, 2,..., n) n n S n =f(x )(x x ) + f(x 2 )(x 2 x ) + + f(x n )(x n x n ) n = f(x k )(x k x k ) k= S = lim n k= S b n f(x k )(x k x k ) b f(x) dx f(x) dx f b integrl n lim f(x k ) (x k x k ) n k= b f(x) S sum dx S 6

f(x) f(x) dx n ( ) f(x) = x 2 S = x 2 k 2 dx lim n n n k= n lim f(x k )(x k x k ) b n k=. 2. 3. b b f(x) dx = f(x) dx = c f(x) dx = f(x) dx+ b f(x) dx b c f(x) dx 2 b b f(x) f(x) f(x) x x f(x) dx f(x) f(x) dx f(x) f(x) F (x) f(x) F (x) = f(x) b f(x) dx = F (b) F () 2. F (x) = x f(x) dx f(x) 2. f(x) F (x) b f(x) dx = F (b) F ().2 f(x) F (x) = x F (x) = f(x) 7 f(x) dx

6. f(x) = x [, 5] F (x) =. F (), F (), F (2), F (3), F (4), F (5) 2. x 5 F (x) = 3. F (x) F (x) = x 7. f(x) = 2x [, 5] F (x) = x. F (), F (), F (2), F (3), F (4), F (5) 2. x 5 F (x) = 3. F (x) F (x) = x x dx x dx = x 2x dx 2x dx = 8. f(x) = 2 x [, 5] F (x) = x. F (), F (), F (2), F (3), F (4), F (5) 2. x 5 F (x) = 3. F (x) F (x) = x 2 2 x dx = ( x 5) x x ( x 5) 2x x x dx x dx ( x 5) 2 x 9 (). [, b] f x b F (x) = x f(x) dx F F (x) = f(x) f(x) F (x) x f(t) dt F (x) 2x dx x 2 x dx 8

Proof. F (x) = x f(x) dx F (x) x f(x) f(x) S = F (x) S x x+h x f(x) dx x+h S f(x) F (x) S = F (x + h) F (x) S F (x) h f(x) h f(x + h) f(x) h < S < f(x + h) h h h > f(x) < S h < f(x + h) S = F (x + h) F (x) h f(x) < F (x + h) F (x) h < f(x + h) F (x + h) F (x) lim f(x) < lim < lim f(x + h) h h h h F (x + h) F (x) lim f(x) = f(x) lim h h h F (x) = f(x) = F (x) lim h f(x + h) = f(x) Remrk. x f(x) dx x x f(x)dx x x 2 x t x f(t) dt x x b t t x 9

.3 f(x). f(x) f(x) F (x) = f(x) F (x) f(x) 2 ( 2). F (x) f(x) b f(x) dx = F (b) F () Proof. G(x) = x f(x) dx 9G (x) = f(x) F (x) f(x) F (x) = f(x) G(x) F (x) (G(x) F (x)) = G (x) F (x) = f(x) f(x) = C x G() = x b G(x) F (x) = C G() F () = C f(x) dx = C = F () b G(x) F (x) = F () f(x) dx = G(b) = F (b) F () F (b) F () [ ] b F (x) F (x) b b [ ] b f(t) dt = F (x)

3.. 2. x 2 dx x 2 F (x) x 2 [ ] F (x) F () F () 2 x 2 dx f(x).4 ( ) 3 x3 = x 2 ( ) 3 x3 x 2 3 x3 + 3 = x 2 ( 3 x3 + 3 x 2 3 x3 + ) 2 = x 2 3 x3 + 2 x 2 x 2 4. f(x) f(x) f(x)dx f(x) dx f(x) f(x) x 2 C 3 x3 + C x 2 x 2 dx = 3 x3 + C 5. f(x) F (x) f(x) C F (x) + C C = F (x) f(x)dx = F (x) + C C f f

f(x) f(x) dx x dx =? x x x dx x x dx? (?)? x (?) = x? 2 x2 x x x x ( ) 2 x2 + C x dx 2 x2 + C x dx = 2 x2 + C f(x).4. x n x dx x 2 dx x 2 x 2 dx (?)? x n dx x 2 dx =? (?) = x 2? = 3 x3 + C x 2 dx = 3 x3 + C x n dx =? (?) = x n? = n + xn+ + C 2

6. x n dx = n + xn+ + C (n =,, 2,... ) dx = x + C dx dx 7. C. x 4 dx 3. x 6 dx 5. x dx 2. x 5 dx 4. x 99 dx.4.2 8.. kf(x) dx = k f(x) dx k 2. 3. ( f(x) + g(x)) dx = ( f(x) g(x)) dx = f(x) dx + f(x) dx g(x) dx g(x) dx Proof. 2. ( f(x) + g(x) (f(x) + g(x)) dx (?)? f(x) dx + ( g(x) dx) = ( f(x) dx) + (f(x) + g(x)) dx =? g(x) dx) = f(x) + g(x) (?) = f(x) + g(x)? = f(x) dx + g(x) dx (f(x) + g(x)) dx = f(x) dx + g(x) dx 9. 3

2. (8x 3 5x + 3) dx = 8 = 8 x 3 dx 5 4 x4 5 x dx + 3 = 2x 4 5 2 x2 + 3x + C dx 2 x2 + 3x + C 2.. (3x 2 4x 2) dx 2. (2x + x 3 ) dx 3. (x )(2x 3) dx 4. t 3 (t 6) dt 5. (2y + 5) 2 dy 6. 5 dx 7. 7 dy 8. 9... 2. 3. 4. (2x 3 3x 2 ) dx (x 3)(2x + ) dx (3x + 2) dx (x 2 6x + 5) dx (8x 3 + 2x 2 + 4x) dx (x + 2)(x 2) dx (3x + 7)(x ) dx, 2, 3 22. 2 F (x). F (x) = x 2 4x, F (3) = 4 () F (x) x 2 4x F (x) = (x 2 4x) dx = (b) F (3) = 4 C C = (c) F (x) = 2. F (x) = 6x 2 4x 5, F (2) = 2 3. F (x) = 3 x, F () = 2 3 4. F (x) = 5x 2 2x + 6, F ( ) = 9 5. F (x) = (3x )( x), F () = 3 4,5 f(x) F (x) f(x) dx F (x) 4

23.. y = f(x) (, ) (x, f(x)) 3(x 2 ) f(x) 2. y = f(x) (, 2) (2, ) (x, f(x)) x 3 x f(x).4.3 x α 24. α x α dx = xα+ α + + C f(x) dx dx f(x) Remrk 25. α = α = x dx 26.. 2. 3. dx x 3 dx x 2 dx x 4. 5. 6. x 6 dx 3 x dx x x dx 7. 8. 6x 4 dx dx x n n n 2 27.. ( x 2 x 3 ) dx 3. (x x) 2 dx 2. ( x + x ) 2 dx 4. x 3 2x 2 dx x 5 5

.5 28. f(x) b F (x) b f(x) dx = 29. f(x) dx f(x) [ ] b F (x) = F (b) F (). 4 x 3 dx x 3 dx = F (x) F (4) F () 2. 3. 2 3 3 (y 2 2y 3)dy (x 3) 2 dx < 6

3.. 9 4 dx x. 3 2 x 2 dx 5. 3 (t 3) 2 dt 2. 3 x 2 dx 2. 5 2 3dx 6. 3 (y 3 4y)dy 3. 3 x 3 dx 3. (2x 2 + 3x + 5)dx 3. β α (x α)(x β) dx = 6 (β α)3 32..6 33. f(x) f( x) = f(x) f( x) = f(x) 34. f(x) f(x) f(x)dx = f(x)dx = 2 f(x)dx 3 4 5 6 y 7

35.. 2. 2 2 2 2 x 2 dx x 4 dx 3. 4. 3 3 (x 3 + 2x 2 + ) dx (4x 3 + 6x 2 9x ) dx 5. 3 3 (x 3) 2 dx 2 2. 36. [, b] f(x) y = f(x) x 2 x =, x = b S f(x) S = b f(x) S = f(x) dx b f(x) dx f(x) S = b f(x) dx f(x) + 37. y = x 3 3x 2 + 2x x S Proof.. x y = x 3 3x 2 +2x = x(x 2)(x ) y = x =,, 2 2. S = (x 3 3x 2 + 2x) dx 2 (x 3 3x 2 + 2x) dx = 2 38. x S 8

. y = 3x 2 x 3 2. y = x 3 + x 2 2x 3. y = x 2 5x + 4 3 4. y = x 2 x 6 3 39. x 2 x =, x = 4. y = 4x x 2 2. y = x 3 9

4. [, b] f(x) g(x) 2 y = f(x), y = g(x) 2 x =, x = b S S = b ( ) f(x) g(x) dx. g(x) b f(x) g(x) S = = b b f(x) dx b g(x) dx ( ) f(x) g(x) dx 2. f(x), g(x) y m f(x) + m g(x) + m f(x) g(x) S = = = b b b (f(x) + m) dx b (g(x) + m) dx ( ) f(x) + m (g(x) + m) dx ( ) f(x) g(x) dx b 4. y = x 2 + 2x + 4 y = x 2. 2 3 x 2 + 2x + 4 = x 2 2x 2 + 2x + 4 = 2(x + )(x 2) = 2. 3 S = 2 2(x + )(x 2) dx = 2 42. 2 (x + )(x 2) dx = 2 6 (2 + )3 = 9. y = x 2 4 y = x 2 2. y = 4x x 2 y = x 4 3. y = x 2 3x 4 y = 2 + x x 2 4. y = 2x 2 7x + 9 y = 5x x 2 2

43.. y = x 2 3x + 5 y = 3 3. y = 2x 2 +3x 5 y = x 2 +4x 3 2. y = 2x 2 + 3x 5 y = x 4. y = x 3 x 2 y = x 44.. y = x 2 x 2. y = x 3 x y 4. y = x 2 + 2x + 3 x 5. y = x 2 y = x 2 + 3x + 5 x = x = 2 3. y = x 2 x 3 x 45. x x =, x = 2. y = x 2 2x + 2 2. y = x 2 + 5x 6 2.2 46. 2 f(x) = x + b f(x) dx =, xf(x) dx = 47. 2 f(x) = x 2 + bx +. f () = 4, 2. f(2) = 3, f(x) dx = xf(x) dx = 3 n lim f(x k ) (x k x k ) n k= b f(x) dx 2 2

(km/ ).5 5 ( ) km/ 5.5km/.5km km/ 5 5.5km/ km/ km 5 5.5 5 = 7.5 5 7.5 km 5 5 f() 2 f(2) 3 f(3) 4 f(4) 5 f(5) = 7.5 (km/ ) 5.5 5 ( ) 5 f(x) dx km/h h = km kg/h h = kg /kg kg = / = / = / = 22

48. 2 f(x) = x 2 3x + 25. 2. 6 2 49. x kg kg y y = 5 4x + 6x 2 3 kg 7 kg 4 kg 5. 3 t y = 4 3 x 4 9 x2 5 5. kg 3 x kg 3x 2. x kg 2..4 kg 3. 2.3 kg 4. x kg 52. x kg kg /kg y = x 2 6x + 2 5 kg 23

[],, [2] 4, 5, IIIC+α [3] I II III,,, [4], [5], [6], S.,, [7], [8] ei =-,, ; (2/) [9], [], [], [2], [3], [4] mth stories,, (29//) [5]!!, [6] :,,, ; (23/3/9) [7],,,,, [8], A.C., K. [9] mth stories, [2], [2],, 24

[22], [23] 96, [24], [25],, (23/5) [26]!,, (2/7/6) [27],,, [28], [29],, [], [2], [3] [4], A.C., K. [5], [6], [7], [8], [9] 4 5 6, [], [] 3 3, [2], [3] :, 25

2.3 (f(g(x))) = f (g(x)) g (x) f(x + b) dx =? (?) = f(x + b)? f(x + b) f(x + b) dx (?)? F (t) f(t) F (t) = f(t) ( ) F (x + b) = F (x + b) = f(x + b)? = F (x + b) + C 53. F (t) f(t) f(x + b) dx = F (x + b) + C f(t) t x+b (x+b) α sin(x + b) 54. (2x ) dx Proof. t = 2x f(t) = t F (t) = (2x ) dx = 2 (2x ) + C = 22 (2x ) + C t dt = t + C f(x + b) 2 x + b t 26

55.. (2x + ) 2 2. (5 4x) 3 3. (x + 3) 2 4. (3x 2) 4 5. (4 3x) 3 6. 7. (2x + 3) 2 3x + 8. (3x 2) 2 9. ( 4x) 3. ( ) 2 2 3 x +. (x + 2) 2. ( 5x) 3 3. 6 ( 2x) 4 4. 4 x 5. 6. 7. 56. Proof. (2x ) dx (2x ) dx = [ 22 (2x ) 2x + 3 ( x 3 + 8 ) 3 8x + 7 ] = 57.. 2. 3. 4. 5. 6. 7. 3 2 4 2 3 2 2 5 (x 3) 2 dx (4 x) 3 dx (2x ) 3 dx (3 2x) dx (2x ) 3 dx 5 2x dx + 3x dx 8. 9... 2. 3. 2 3 2 2 2 5 2 3 2 (2x 3) 4 dx dx 3x 5 5 2x dx + 3x dx (2x 3) 4 dx dx 3x 5 27

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