x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n

1, R f : R R,.,, b R < b, f(x) [, b] f(x)dx,, [, b] f(x) x ( ) ( 1 ). y y f(x) f(x)dx b x 1: f(x)dx, [, b] f(x) x ( ).,,,,,., f(x)dx,,,, f(x)dx. 1.1 Riemnn,, [, b] f(x) x., x 0 < x 1 < x 2 < < x n 1 < x n b (1) 1

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n [, b],,, {x 0, x 1, x 2,, x n }. 1,, γ i [x i 1, x i ], γ (γ 1, γ 2,, γ n )., [x i 1, x i ], f(γ i ),, Riemnn S( ; γ) f(γ i )(x i x i 1 ) i1, S( ; γ) γ f(x) Riemnn ( 3 )., [x i 1, x i ], x i x i x i 1, Riemnn S( ; γ), Riemnn ( ) S( ; γ) f(γ i ) x i (2) i1, ( ). 1,, n + 1. 2

y y f(x) S( ; γ) x 0 x 1 x 2 x 3 x n 1 x n x γ 1 γ 2 γ 3 γ n 3: γ i [x i 1, x i ], Riemnn S( ; γ),. 1.2, 1.1, [, b] γ, x i f(γ i ),, Riemnn S( ; γ)., [, b] f(x) x,,,.,, 2,.,, f(x)dx, Riemnn.,,.,,, mx 1 i n x i x i 1 mx 1 i n x i,.,, f(x),, [, b], Riemnn S( ; γ),,., lim S( ; γ) (3) 0,.,,, [, b] f(x) x. 2,, n + 1. 3

,,,, (3)., (3), f(x) [, b],, f(x)dx lim S( ; γ) (4) 0., (2) Riemnn, (4), ( ) f(x)dx lim f(γ i ) x i (5) 0 i1, f(x)dx. 3, (2) Riemnn, f(x), f(x),,, lim S( ; γ) (6) 0, (6),.,,, f(x),, [, b] f(x) x, (6).,,. 4,, Riemnn S( ; γ),, f(x) x, (5).,, (6),,,,. 3, R, (sum) P,., S Σ. 4 IB 10, f(x).,. 4

2 2.1, 1, f(x) f(x)dx, f(x)dx lim S( ; γ) (7) 0, Riemnn S( ; γ)., f(x), (7),.,,,.,, [, b],, b, F (b) f(x)dx, F (b) b.,,, b x, F (x) x F (x) f(t)dt (8), x. 5, x 0 R,, h R, F (x 0 + h) F (x 0 )., F (x 0 + h) F (x 0 ),, 4, F (x 0 + h) F (x 0 ), F (x 0 + h) F (x 0 ) x0 +h x 0 f(x)dx ( 5 )., h < 0,. 6 x0 +h x 0 f(x)dx x0 x 0 +h f(x)dx 5, x,, x t. 6,, > b, [, b], [b, ], Z b f(x)dx Z b f(x)dx.,,, [, b]. 5

y F (x 0 ) y f(t) F (x 0 + h) x 0 x 0 + h t 4: F (x 0 + h) F (x 0 ),,. F (x 0 + h) F (x 0 ) x 0 +h x 0 f(x)dx y y f(t) f(x 0 ) x 0 x 0 + h t h 5: F (x 0 + h) F (x 0 ),.,, f(t) R.,, f(t) t x 0, t x 0, f(t) f(x 0 )., h, [x 0, x 0 + h] t [x 0, x 0 + h], (, h < 0, [x 0 + h, x 0 ] t [x 0 + h, x 0 ], ) t x 0, t [x 0, x 0 + h], (, h < 0, t [x 0 + h, x 0 ], ) f(t) f(x 0 ) (9)., (9), x0 +h x 0 f(t)dt x0 +h x 0 f(x 0 )dt (10), h, F (x 0 + h) F (x 0 ) x0 +h x 0 f(t)dt 6

x0 +h f(x 0 )dt ( (10) ) x 0 f(x 0 ) x0 +h x 0 dt f(x 0 ) h (11). 7 h 0,, (11) h, df dx (x F (x 0 + h) F (x 0 ) 0) lim h 0 h f(x 0 ) (12). 8, x 0 R,, x 0 x, F (x) f(x) df (x) f(x) (13) dx., F (x) f(x)., f(x), f(x) dg (x) f(x) (14) dx G(x) f(x)., (13), F (x) x f(t)dt, f(x) ( ).,, (14) f(x) G(x).,, F (x) G(x),, F (x)., F (x) G(x), (13), (14), d df dg (F (x) G(x)) (x) dx dx dx (x) f(x) f(x) ( (13), (14) ) 7, h, F (x 0 + h) F (x 0 ) R x 0 +h x 0 f(t)dt, h f(x 0 ) ( 5 ). 8, IB 10. 7

0., C R,., F (x) G(x) C (15) F () f(t)dt 0, (15), x, C F () G()., (15), (16), F (x) G(x) + C G() (16) ( (15) ) G(x) G() ( (16) )., x f(t)dt G(x) G() (17).,, x b, t x,, f(x)dx G(b) G() (18)., f(x) G(x),, G(x), f(x) f(x)dx.9, (18), G(b) G() [G(x)] b, f(x)dx [G(x)] b G(b) G(), (18), f(x) G(x) 9, f(x),.,. 8

,, G(x), f(x) f(x)dx., f(x) f(x). 2.2, (x) 1, f(x) 1, f(x), G(x) x, 1dx [x] b b (19). 10, (x 2 ) 2x, f(x) x, f(x), G(x) x2 2, [ x 2 xdx 2 ] b b2 2 2 2 (20). 11, n N, (x n+1 ) (n + 1)x n, f(x) x n, f(x), G(x) xn+1 n+1, 10, [, b] f(x) x, (19), (b ) 1 (b ). 11,, [, b] f(x) x, (20),,, b, (b ),. b 2 2 2 2 (b + )(b ) 2 9

x n [ x x n n+1 dx n + 1 ] b bn+1 n + 1 n+1 n + 1., sin x, cos x, (sin x) cos x (cos x) sin x (21), f(x) sin x, f(x), G(x) cos x,, f(x) cos x, f(x), G(x) sin x, sin xdx [ cos x] b cos b + cos (22) cos xdx [sin x] b., π sin b sin (23) 0 sin xdx [ cos x] π 0 cos π + cos 0 ( 1) + 1 2, sin 2., e x, (e x ) e x, f(x) e x, f(x), G(x) e x, ( ) e x e x dx [e x ] b e b e (24) 10

. 2.3, g(x), h(x), f(x) g(x) + h(x) (25)., [, b] γ,, f(x) Riemnn S f ( ; γ), 12 S f ( ; γ) f(γ i ) x i i1 (g(γ i ) + h(γ i )) x i i1 g(γ i ) x i + i1 h(γ i ) x i i1 S g ( ; γ) + S h ( ; γ) (26)., (26), 0, (g(x) + h(x))dx g(x)dx + h(x)dx (27)., g(x), h(x),, G(x), H(x), (27),, (g(x) + h(x))dx g(x)dx + [G(x)] b + [H(x)] b h(x)dx (G(b) G()) + (H(b) H()) (G(b) + H(b)) (G() + H()) [G(x) + H(x)] b 12, Riemnn,, 1, γ f(x) Riemnn, S( ; γ), S f ( ; γ), f. 11

( ) (g(x) + h(x))dx [G(x) + H(x)] b (28). 13, 2.2, [ ] x (x + x 3 2 b )dx 2 + x4 ((21), (28) ) 4 ( ) ( ) b 2 2 + b4 2 4 2 + 4 4., f(x), f(x) g 1 (x) + g 2 (x) + g 1 (x),,,, (27), (g 1 (x) + g 2 (x) + g 3 (x)) dx f(x) {g 1 (x) + g 2 (x)} + g 3 (x) ({g 1 (x) + g 2 (x)} + g 3 (x)) dx {g 1 (x) + g 2 (x)}dx + g 3 (x)dx { } g 1 (x)dx + g 2 (x)dx + g 3 (x)dx g 1 (x)dx + g 2 (x)dx + g 3 (x)dx ((27) ) ((27) ).,, n N, f(x) n, ( ) (g 1 (x)+g 2 (x)+ +g n (x))dx 13 (28), g 1 (x)dx+ g 2 (x)dx+ + g n (x)dx (29) (G(x) + H(x)) G (x) + H (x) g(x) + h(x), f(x) g(x) + h(x) F (x), F (x) G(x) + H(x). 12

. 14, g i (x) G i (x), (29), (, ) (g 1 (x) + g 2 (x) + + g n (x))dx [G 1 (x) + G 2 (x) + + G n (x)] b (30). 15 2.4, g(x), f(x) 2g(x)., [, b] γ,, f(x) Riemnn S f ( ; γ), S f ( ; γ) f(γ i ) x i i1 2g(γ i ) x i i1 2 g(γ i ) x i i1 2S g ( ; γ) (31)., (31), 0, ( ) 2g(x)dx 2 g(x)dx (32)., C R, 14, n, (29). 15, (30), (G 1 (x) + G 2 (x) + + G n (x)) G 1(x) + G 2(x) + + G n(x) g 1 (x) + g 2 (x) + + g n (x), f(x) g 1 (x) + g 2 (x) + + g n (x) F (x), F (x) G 1 (x) + G 2(x) + + G n(x). 13

Cg(x)dx C g(x)dx (33)., g(x) G(x), (33), Cg(x)dx C g(x)dx C [G(x)] b C(G(b) G()) CG(b) CG() [CG(x)] b, ( ) Cg(x)dx [CG(x)] b (34). 16, 2.3, C 1, C 2,, C n R, f(x), f(x) C 1 g 1 (x) + C 2 g 2 (x) + + C n g n (x), n, 17 g i (x) G i (x), (29), (34), (C 1 g 1 (x) + C 2 g 2 (x) + + C n g n (x))dx 16 (34), C 1 g 1 (x)dx + C 2 g 2 (x)dx + + C n g n (x)dx ((29) ) [C 1 G 1 (x)] b + [C 2 G 2 (x)] b + + [C n G n (x)] b ((34) ) (C 1 G 1 (b) C 1 G 1 ()) + (C 2 G 2 (b) C 2 G 2 ()) + + (C n G n (b) C n G n ()) (C 1 G 1 (b) + C 2 G 2 (b) + + C n G n (b)) (C 1 G 1 () + C 2 G 2 () + + C n G n ()) (CG(x)) CG (x) Cg(x), f(x) Cg(x) F (x), F (x) CG(x). 17,. 14

[C 1 G 1 (x) + C 2 G 2 (x) + + C n G n (x)] b, ( ) (C 1 g 1 (x)+c 2 g 2 (x)+ +C n g n (x))dx [C 1 G 1 (x)+c 2 G 2 (x)+ +C n G n (x)] b. 18, 2.2, ] b (3x 2 + 5x + 1)dx [3 x3 3 + 5 x2 2 + x [ x 3 + 5 ] b 2 x2 + x ( b 3 + 5 ) ( 2 b2 + b 3 + 5 ) 2 2 + (35) ((21), (35) ), (2 cos x + 5e x )dx [2 sin x + 5 e x ] b ((23), (24), (35) ) [2 sin x + 5e x ] b ( 2 sin b + 5e b) (2 sin + 5e ). 2.5, g(x), h(x),, f(x) g(x)h(x), f(x) g(x)h(x) (g(x)h(x)) g (x)h(x) + g(x)h (x) (36),.,, b R, (36) b, 18 (35), (g(x)h(x)) dx g (x)h(x)dx + g(x)h (x)dx (37) (C 1G 1(x) + C 2G 2(x) + + C ng n(x)) C 1G 1(x) + C 2G 2(x) + + C ng n(x) C 1 g 1 (x) + C 2 g 2 (x) + + C n g n (x), f(x) C 1 g 1 (x) + C 2 g 2 (x) + + C n g n (x) F (x), F (x) C 1G 1(x) + C 2G 2(x) + + C ng n(x). 15

., (g(x)h(x)) dx [g(x)h(x)] b (38), (37), g (x)h(x)dx [g(x)h(x)] b g(x)h (x)dx (39), (39)., (39), g (x)h(x)dx [g(x)h(x)] b (40), (40), g (x)h(x), g (x) ( ), g(x)h(x)., (39).,, g(x)h(x), g (x)h(x), g (x)h(x) + g(x)h (x), (40), (39),., (39), g 0 (x)h(x) g(x)h 0 (x)., g(x), h(x),, h(x) h 0 (x)., h(x) h(x) h 0 (x),,., x sin xdx,, x sin x., x sin x, x, x (x) 1, x sin x sin x,. 19,, g (x) sin x, h(x) x 19, g (x)h(x) g (x)h (x), g (x)h(x) g(x)h (x),, x sin x cos x. 16

, (39), x sin xdx x ( cos x) dx [x ( cos x)] b [ x cos x] b [ x cos x] b + (x) ( cos x)dx ( cos x)dx ( (39) ) cos xdx (41), 20 (41), x sin x cos x., cos x, (41), x sin xdx [ x cos x] b + [ x cos x] b + [sin x] b [ x cos x + sin x] b cos xdx., f(x) x sin x, F (x) x cos x + sin x.,,, log x dx (log x) 1 x,, log x., log x, log x (log x) 1 x,.,, 1, log x 1 log x, g 0 (x) 1, h(x) log x, (39)., log x dx 1 log x dx (x) log x dx [x log x] b x (log x) dx 20, g (x) sin x, g(x) cos x. ( (39) ) 17

[x log x] b [x log x] b x 1 x dx 1dx (42), 21 (42), log x 1., 1, (42), log x dx [x log x] b [x log x] b [x] b [x log x x] b., f(x) log x, F (x) x log x x.,,,. 2.1, f(x), f(x) F (x), 1dx f(x)dx [F (x)] b (43) F (b) F (), f(x). 22, (43),,, b.,, (43), b, b, f(x)dx F (x) (44)., f(x)dx, f(x). f(x)dx, f(x)dx f(x). 23 21, g (x) 1, g(x) x. 22 2.1, F (x) R x f(t)dt, f(x) G(x),, f(x) F (x). 23 [, b]., R b f(x)dx, [, b], R b f(x)dx. 18

,,, log x dx x log x 1dx x log x x,., 2.1, f(x), f(x),,., (44) f(x)dx, f(x),.,,, 1 x dx 1 1 x dx (x) 1 x dx x 1 ( ) 1 x x dx (. ) x 1 x ( 1x ) 2 dx 1 1 + dx (45) x,, (45) 1 x dx, 0 1 (46),., 1 x dx, (46), 0 1,.,,,,,, 1 x dx, 1 x x dx 1 0 t dt, 1 x dx,,., (45), b, 1 b x dx 1 [1]b + x dx 19

1 (1 1) + x dx 1 x dx,. 2.6, 1, f(x), [, b] f(x), f(x)dx lim S( ; γ) (47) 0, Riemnn.,, x, x x(t), t, (47), t,.,,., R ϕ : R R,, β,, b R,, < β, < b, ϕ, [, β] [, b]. 24, [, β] t [, β], ϕ(t) [, b],, [, b] x [, b], x ϕ(t) [, β] t [, β] ( 6 )., [, b] t [, β]. 25, ϕ(t) [, β],,,, ϕ(t). 26 24, x x(t), x,,,, x(t), ϕ(t). 25, ϕ R, ϕ : [, β] R, [, β],, [, β],, ϕ R. 26 ϕ(t),., ϕ(t),,, x ϕ(t), [, β] [, b],. 20

x x ϕ(t) b β t 6: ϕ, [, β] [, b].,, t 0 < t 1 < t 2 < < t n 1 < t n β [, β] {t 0, t 1, t 2,, t n } (48), ξ i [t i 1, t i ] ξ (ξ 1, ξ 2,, ξ n ) (49),., i 0, 1, 2,, n, x i ϕ(t i ) (50), x 0, x 1, x 2,, x n, {x 0, x 1, x 2,, x n } (51) [, b]. 27, i 0, 1, 2,, n, γ i ϕ(ξ i ) (52), γ 1, γ 2,, γ n, γ {γ 1, γ 2,, γ n } (53) [, b]. 28 27,., t, x ϕ(t), t x, x, t β x b., t t i x i. 28, t, t ξ i γ i. 21

, [, b] γ f(x) Riemnn S f ( ; γ) f(γ i ) x i (54) i1. 29, i 0, 1, 2,, n, x i x i x i 1 (55)., (54) Riemnn S f ( ; γ) t., (50), (52), (55), S f ( ; γ) f(ϕ(ξ i ))(ϕ(t i ) ϕ(t i 1 )) (56) i1.,, i 0, 1, 2,, n, t i t i t i 1, ϕ(t i ) ϕ(t i 1 ) ϕ (η i ) t i (57) η i R t i t i 1 ( 7 ), (56), (57), S f ( ; γ) f(ϕ(ξ i ))ϕ (η i ) t i (58) i1. x x ϕ(t) x i x i 1 x i ϕ(t i ) ϕ(t i 1 ) t i 1 η i ti t t i 7: ϕ(t i) ϕ(t i 1 ) t i ϕ (η i ) η i R t i t i 1., (58), g(t) f(ϕ(t))ϕ (t) (59) 29, Riemnn,, 1, γ f(x) Riemnn, S( ; γ), S f ( ; γ), f. 22

, g(t) Riemnn, f(ϕ(t)) f(ϕ(ξ i )) t ξ i, ϕ (t) ϕ (η i ) t η i,, g(t) Riemnn.,, (57) η i η i [t i 1, t i ], η {η 1, η 2,, η n } (60), ξ η., Riemnn,, [, β], ξ, ξ, (60), 0 ξ η.,,, (58), ξ i η i (61) S f ( ; γ) f(ϕ(ξ i ))ϕ (η i ) t i i1 f(ϕ(η i ))ϕ (η i ) t i i1 g(η i ) t i i1 S g ( ; η) ( (61) ) ( (59) ).,, ( )Riemnn S f ( ; γ) S g ( ; η) (62) f(x) Riemnn g(t) Riemnn., (62), 0., 0, 0, lim S f ( ; γ) lim S g( ; η) (63) 0 0 23

.,, lim S f ( ; γ) 0 lim S g( ; η) 0 β β f(x)dx (64) g(t)dt f(ϕ(t))ϕ (t)dt (65),, (63), (64), (65), β f(x)dx f(ϕ(t))ϕ (t)dt (66). (66)., ϕ(t) x(t), (66), f(x)dx β f(x(t))x (t)dt,, dx(t) x (t)dt (67), (66), ( ) β f(x)dx f(x(t))dx(t) (68) β f(x(t))x (t)dt (69)., (68), x x(t), x t, dx x, x x x(t),, (69).,, (57), x i x(t i ) x(t i 1 ) x (η i ) t i (70), (70), (67)., (70), x i t i x (η i ) 24

, t i 0, (67),, dx(t) dt x 0 (t) (71), dt., f(x) F (x), f(x)dx [F (x)] b F (b) F () (72),, b,, (72), (73), x(), b x(β) (73) f(x)dx F (b) F () F (x(β)) F (x()) [F (x(t))] β (74)., (74), (69), β f(x(t))x (t)dt [F (x(t))] β (75). (75), f(x(t))x (t), F (x(t)),, (69), F (x(t)), d df dx F (x(t)) (x(t)) dt dx dt (t) f(x(t)) x (t).,,, ( ) f(x) f(x(t))x (t). ( ) f(x(t))x (t) f(x). 25

. 30,, dx 1 + x 2 (76) x tn θ (77), ( ). 31, (76), 1 1 + x 2 1 1 + tn 2 θ 1 1 + sin2 θ cos 2 θ cos 2 θ cos 2 θ + sin 2 θ cos 2 θ (78)., (77) θ,, (79), dx dθ 1 cos 2 θ. 32, (78), (80), dx 1 + x 2 (79) dx 1 cos 2 dθ (80) θ β β [θ] β cos 2 θ dθ 1 cos 2 θ dθ [tn 1 x] b (81)., x,, x b θ, θ,, θ β. 33 (81), f(x) 1 1+x 2, F (x) tn 1 x tn,, (tn 1 x) 1 1 + x 2 30, ( ), x x(t), x t, t t(x), t x, ( ) ( ). 31, x x(t) tn t, tn t t,, t θ, x x(θ) tn θ, x θ. 32, (67), (71), dt. 33, tn, b tn β,, β ( π, π ). 2 2 26

., β dt e t + e t (82)., (82),, x e t (83), 1 e t + e t 1 x + x 1 1 x + 1 x x x 2 + 1 (84),.,,, x x(t), t x, ( )., (83) t,, (85), dt, dx dt et (85) dx e t dt (86)., (86) e t, dt 1 e t dx 1 dx (87) x. 34, (84), (87), β dt b e t + e t x x 2 + 1 1 dx (88) x dx x 2 + 1 (89) 34,, (82) x, (86) dx t, (87) dt x. 27

. 35, t,, t β x, x,, x b. 36, (81) (89), β dt b e t + e t dx x 2 + 1 [tn 1 x] b.,, dx 1 + x 2, [tn 1 (e t )] β x tn θ (90), dx, θ, dx β 1 + x 2 dθ.,, β dt e t + e t x e t (91), dt, x, β dt b e t + e t dx x 2 + 1.,, ( ), ( ),,, (90), 35, (87), (88),,, Z β Z dt β e t + e 1 t e t + e 1 t e t et dt, 1 f(x(t)) e t + e 1 t e, t, (69). 36, e, b e β. x (t) e t 28

(91),, d,, ( ), ( ),. 29