. () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x (x )(x ) dx ( t dt +t + ) dt t t + log x + x + C log + C t x x sin x t t, cos x +t +t, dx dt +t tdt t
+ sin x +t + t sin x( + cos x) dx dt t ( t + t + ) dt log t + 4 t + t + C log tn x + x 4 tn + tn x + C. () ( + x)( + x ) dx () + cos x dx x +x dx +x 4 ( () 3 x + + ( + x) log 6 x + x + rctn 3 () rctn x + C ( ) x rcsin x + + C ( x 3 ) + C. () () π + cos x dx x x +4 dx log( + x)dx () t tn x cos x t +t, dx dt +t π + cos x dx t +3 dt
() [ [ 3 rctn t ] π 3 3 3 x x +4 dx ( ) x 4 +4 dx x +4 x x +4+4log ( x + x +4 )} 4 log ( x + x +4 )] [ x x +4 log ( x + x +4 )] 5 log + 5 t tn x x t, dx tdt log( + x)dx t log( + t)dt [ t log( + t) ] +t dt ( log t + ) dt t + log [ t t t + log t + ]. () () π π 4 sin x + cos x dx ( + x ) dx log( + tn x)dx 3
() cos x t π () x tn t sin x + cos x dx x π 4 t I I π 8 log π ( + x ) dx 4 π 4 ( log +t dt [log( + t)] log π 4 + tn t cos t dt π 4 + ) dt π 4 log I. () (x + y )dxdy, D y, y x, x D () xyz(x + y + z)dxdydz, D 3 D 3 x + y + z, x, y, z (x + x + + x n)dx dx dx n, D n D n x i, i,,,n () () D (x + y )dxdy x x y x x } (x + y )dy dx 4 3 x3 dx 3 xyz(x + y + z)dxdydz D 3 } } xyz(x + y + z)dz dy xy x + y + } ( x y) dy dx 6 dx 4
x(x + y x +) x + y +3 ( x y) 6 x 3 (x )5 + x 4 4 (x )4 + x } (x ) 3 dx 6 (x )6 + (x )5 + } (x )4 dx 4 84 (x + x + + x n)dx dx dx n D n } } } (x + x + + x n)dx dx dx n ( 3 + x + + x n ) } } dx dx n n 3. ( ) y () exp dxdy, D y, y x, x D x () (x + y + z )dxdydz, D 3 D 3 x + y + z, x, y, z (x + x + + x n )dx dx dx n, D n D n x i, i,,,n () () D exp ( y x ) dxdy x x y x ( ) } y exp dy dx e x (x + y + z )dxdydz D 3 } (x + y + z )dz (x + x + + x n )dx dx dx n D n } dy dx 5
} } (x + x + + x n )dx dx ( ) } } + x + + x n dx dx n n } dx n. () x dxdy, D x + y D () D x y dxdy, D y mxx, x} x dxdydz, D 3 x + y + z D 3 ) () () x r cos θ y r sin θ, dxdy rdrdθ r<, θ π π ( ) x dxdy r 3 cos 3 θdr dθ D π cos θdθ π + cos θ dθ π 4 4 4 X x + y Y x y, dxdy dxdy X, Y X ( ) X D x y dxdy XY dy dx dx x r sin θ cos ϕ y r sin θ sin ϕ z r cos θ 6
, dxdydz r sin θdrdθdϕ r, ϕ π, θ π π π ( ) } x dxdydz r 4 sin 3 θ cos ϕdr dϕ dθ 4 D 3 5 π. () (x + y ) 3 dxdy, D x + y x D ( ) y x () exp dxdy, D x, y, x+ y D y + x x dxdydz, D 3 x + y + z D 3 () (x ) + y x r cos θ y r sin θ, dxdy rdrdθ r< cos θ, π θ π D (x + y ) 3 dxdy π π ( ) cos θ r 4 dr dθ 5 75 () X x + y Y x y, dxdy dxdy X, X Y X D exp ( ) y x dxdy y + x 4 X X ) ( e e ( Y exp X ) } dy dx 7
x r cos θ y r sin θ z z, dxdydz rdrdθdz r + z, z x dxdydz D 3 x dxdydz x +y +z, z r +z, z π ( r r 3 cos θdrdθdz r 3 cos θdz ) dr } dθ π 6 () (). sin x dx x sin x dx x sin x () lim x + x,. ε>, K> t >t >K [ t t sin x x dx cos x x t + t + ] t t t t K <ε cos x x dx t t x dx + + < t t t t t K <ε sin x dx. x 8
() n,,, 3, (n+)π sin x π nπ x dx sin t nπ + t dt > π sin tdt nπ + π (n +)π nπ sin x x dx > π ( + + 3 + + ) n (n ). f :[, ) R, x f(x) x N, e x f(x)dx. N. lim x e x x N+, e x x N+ [, )., M> e x x N+ <M ( x [, )) e x x N < M x ( x (, )). ε> K < ε M K> t >t >K t e x t f(x)dx t t t e x x N dx < M t x dx < ε e x f(x)dx. () () x dx x + dx log xdx. 9
() N < N [, N] x x dx N lim N x dx. lim N [rcsin x]n π () N < N [, N] x +. N dx lim x + N dx lim x + [rctn N x]n π <N N [N, ] log x. log xdx lim log xdx lim [x log x N + N N + x] N () (). dx e x + e x x 3 + dx log x dx ( <α<) xα () t e x () N dx lim e x + e x N x 3 + dx lim N 3 N + x + dx N 6 N ( x ) + 3 4 +t dt π 4 x x x + dx dx 3 3 π
[ ] log x x α dx lim xα N + α log x N ( α) α N x α dx f :[, ) [, ), (i) (ii). (i) f(n) (ii) n f(x)dx f(x), n } n n n f(x)dx (n,, 3, ), (ii) n } n. f(x) f(x) n x n + f(n) f(x) f(n +). f(n) n+ n f(x)dx f(n +) m m f(n) m+ f(n +) n n n } m n f(n)} m. (i). n ζ(s) n n s
s>, s. s n ζ(s). ns f(x), s> f :[, ) (, ) xs, x dx s s> s s< s lim log x x ζ(s), s. R e (x +y) dxdy,. e x dx π. R x r cos θ y r sin θ, dxdy rdrdθ R n D R (r cos θ, r sin θ) r R, θ<π} R lim R D R R π e (x +y) dxdy lim re r drdθ R R R lim π re r dr R n s s> lim R π [ e r] R π
( R e x dx π. R ) e x dx e (x +y) dxdy π R,. () D x + y dxdy, D x y () D x + y dxdy, D x + y () D (x, y) x y } D n (x, y) x y, n y } D lim n D n. dxdy lim D x + y n D n x + y dxdy ( ) y lim n n x + y dx dy n lim log ( + ) ( ) log ( + ) n () D (x, y) x + y } D n (x, y) n x + y } D lim n D n. x r cos θ y r sin θ dxdy lim D x + y n D n x + y dxdy ( π ) n lim r dr dθ n lim n π log n 3
f(x), g(x) [, b],. ( ) b ( b f(x)g(x)dx f(x) ) b dx g(x) dx t R b (tf(x)+g(x)) dx At + Bt + C A b f(x) dx, B b f(x)g(x)dx, C b g(x) dx. A f(x) ( x [, b]),. A> At + Bt + C ( t R), 4B 4AC, B AC B A C. ( b ) ( b f(x)g(x)dx f(x) b dx g(x) dx ) f(x), g(x) [, b], p, q p + q.,. ( b ) ( b f(x)g(x)dx f(x) p p ) b dx g(x) q q dx b f(x) p dx b g(x) q dx f(x)g(x) ( x [, b]). α f(x) ( b f(x) p dx ) p, β g(x) ( b g(x) q dx ) q 4
, αβ p αp + q βq ( α, β ) ( b f(x) p dx ) ( p b g(x) q dx ) q b f(x)g(x) dx p b f(x) p dx p + q b f(x) p dx + q b g(x) q dx b g(x) q dx ( ) b ( b f(x)g(x)dx f(x) p p b dx g(x) q dx ) q. f :[, b] R, f(x) B f B f x [, b] x f(x) }, ε>, R I n } n, I n I n B f I n n I n <ε n. B f ),. () f :[, ] R, f(x) ( x ) ( x ) 5
() f :[, ] R ( x ) f(x) ( x. x p q q ) () f :[, ] R B f B f [, ] B f,. f. () f B f B f x [, ] x }, B f. f.. () f :[, ] R f(x) x ( x ) x ( x ) () f :[, ] R, [, ], q ( ),q,q 3,,q n,. q,q,q 3,,q n, } [, ] } f(x). n f(x) k, ( x q n x<q n+ ) k () f B f x [, ] x }, B f. f. () f B f q,q,q 3,,q n, }, B f. f. 6