() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε > 0 δ > 0 = mx x i x i < δ i ξ i ( ξ i [x i, x i ]) S = f(ξ i )(x i x i ) S < ε, S, b f(x)dx
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
(3) 6. [, b] (Riemnn). n (f) = f(ξ i )(x i x i ) i= = {x, x,, x n }. f(x) ( L f(x) < L) sup f(x) = M i, inf f(x) = m i. x i x x i x i x x i n m i (x i x i ) (f) i= n M i (x i x i ) i= m i f(ξ i ) M i s = S = n m i (x i x i ) i= n M i (x i x i ) i=. (i) s S (m i M i ) (ii) ( ) s s S S. ) = {x, x,, x n }. [x i, x i ] y. m i = inf f(x) x i x x i inf f(x) m i x i x y inf f(x). y x x i m i (x i x i ) inf f(x)(y x i ) + inf f(x)(x i y) x i x y y x x i s [x i, x i ] s [x i, x i ] i s s. S S.
(4) (iii) s S 3 = s s 3 S 3 S s S ( ) S,s inf sup inf S sup s f(x) inf S = sup s ε > 0 S s < ε f(x) [, b]. ε > 0 δ x y < δ f(x) f(y) < ε δ < δ x i x i < δ x, y [x i, x i ] ε < f(x) f(y) < ε [x i, x i ] x sup, y inf ε M i m i ε f(x) f(y) < ε M i m i ε S s ε(b ) inf S = sup s. S.
(5) s (f) S = mx(x i x i ) < δ S s ε(b ) (f) S = S s ε(b ) S (f) S S < S. s S < S S S s ε(b ) S f(x) g(x) [, b] () f(x) + g(x) b (f(x) + g(x)) dx = b f(x)dx + b g(x)dx [ ] f(x) ε > 0 δ > 0 = mx i x i x i < δ f(ξ i )(x i x i ) b f(x)dx < ε g(x) ε > 0 δ > 0 = mx x i x i < δ i g(ξ i )(x i x i ) b g(x)dx < ε δ = min(δ, δ ) (min(δ, δ ) = {δ, δ } δ, δ ) < δ (f(ξ i ) + g(ξ i ))(x i x i ) f + g f(ξ i )(x i x i ) b f + g (f(x) + g(x))dx = b b b f(x)dx b g(x)dx f(x)dx + g(ξ i )(x i x i ) f(x)dx + b g(x)dx b g(x)dx < ε
(6) () c R b cf(x)dx = c b f(x)dx (3) < c < b b f(x)dx = c f(x)dx + b c f(x)dx (4) f(x) 0 b f(x)dx 0 (5) f(x) g(x) b f(x)dx b g(x)dx (6) M = sup f(x), m = inf f(x) x b x b m f(x) M x [, b] [, b] m(b ) b f(x)dx M(b ) p (m p M) b f(x)dx = p(b ) Remrk f(x) [, b] M = sup f(x) = mx f(x) x b x b m = inf f(x) = min f(x) x b x b f(ξ) = p (m p M) ξ [, b]. b f(x)dx = f(ξ)(b )
(7) (7) b b f(x)dx f(x) dx ( f(x) f(x) f(x) ) (8) ( ) b f(x)g(x)dx b f(x) dx b g(x) dx [ ] λ b f(x) + λg(x) dx 0. (9) f(x) g(x) y = f(x), y = g(x), x =, x = b b (g(x) f(x))dx
(8) 6. F (x) = x f(t)dt f(x), f(x)dx 6. f(x) F (x) = f(x) (6) F (x + h) F (x) F (x) = lim h 0 h = lim h 0 = lim h 0 x+h x+h x f(t)dt x f(t)dt h f(t)dt h h f(x + θh) = lim (0 θ ) h 0 h = lim f(x + θh) = f(x) h 0 Remrk F (x) = f(x) f(x). f(x) f(x) C
(9) 6. (6.) x α+ x α + C (α ) dx = α + log x + C (α = ) (6.) cos x dx = sin x + C (6.3) sin x dx = cos x + C (6.4) e x dx = ex + C (6.5) cos x dx = tn x + C (6.6) sin x dx = cot x + C (6.7) ( x ) tn = + ( x ) = x + x + dx = x tn + C (6.8) x dx = sin x + C (6.9) f (x) dx = log f(x) + C f(x)
(0) (6.0) x dx = ( x ) dx = x + log x x + + C (6.) x dx = e x log dx = log ex log + C = log x + C (6.) tn x dx = sin x sin x cos x dx = cos x dx = log cos x + C (6.3) x dx = log x + x + C ( x > ) (6.4) x dx = ( x ) x + sin x + C (6.5) x dx = ( x x log x + ) x + C ( x > ) (6.6) x + dx = ( x x + log x + ) x + C
() F (x) f(x). F (x) = f(x) 6. d dx u(x) c b f(t)dt f(x)dx = [F (x)] b = F (b) F () f(x) F (t) d u(x) f(t)dt = [F (t)] u(x) c dx c Riemnn = d (F (u(x)) F (c)) dx = d dx F (u(x)) = F (u(x)) u (x) = f(u(x)) u (x) 6.3 ( lim + + 3 + + n ) n n 3 Riemnn S n = n 3 ( + + 3 + + n ) = n { ( ) + n S n [0, ] y = x lim S n = n 0 x dx = 3 ( ) } ( n ) + + n n S n y = x [0, ]
() 6.3 G(x) g(x) f(x)g(x)dx = f(x)g(x) f (x)g(x)dx b f(x)g(x)dx = [f(x)g(x)] b b f (x)g(x)dx [ ] f(x)g(x) d dx (f(x)g(x)) = f (x)g(x) + f(x)g (x) = f (x)g(x) + f(x)g(x) b 6.4 () () b [(f(x)g(x))] b = f (x)g(x) + b log xdx = log xdx = (x) log xdx = x log x x x dx = x log x dx = x log x x + C (C: ) ( ) x x(log x) dx = (log x) dx = x (log x) x log xdx ( ) = x x (log x) log xdx ( = x (log x) x x log x + ) dx x = x (log x) x x log x + 4 + C (C: ) f(x)g(x)
(3) 6.4 x = ϕ(t) (ϕ(t) : ) f(x)dx = f(ϕ(t))ϕ (t)dt ϕ(b) f(x)dx = b ϕ() f(ϕ(t))ϕ (t)dt Remrk. f(x)dx = f(x(t)) dx dt dt ( ) [ ] F (x) = x ϕ() f(t)dt. F (x) = f(x). x = ϕ(t) ϕ (t). F (x)ϕ (t) = f(x)ϕ (t) b d dt (F (ϕ(t))) = f(ϕ(t))ϕ (t) F (ϕ(b)) F (ϕ()) = = 6.5 (log x) x dx t = log x dt dx = x (log x) dx = x x t dx dt dt = x t xdt = t dt ϕ(b) ϕ() = 3 t3 + C = 3 (log x)3 + C f(x)dx = b b f(ϕ(t))ϕ (t)dt f(ϕ(t))ϕ (t)dt
(4) () e x 6.6 I = I = (e x ) sin xdx = e x sin x e x cos xdx e x sin xdx ( ) e x cos xdx = e x cos x + e x sin xdx ( ) I = e x sin x e x cos x + e x sin xdx I = e x sin x e x cos x I I = e x sin x e x cos x I = ex (sin x cos x) e x sin xdx = ex (sin x cos x) + C e ix = cos x + i sin x e iπ + = 0 e ix = cos x + i sin x = e x e ix dx = e x (cos x + i sin x)dx e (+i)x dx = + i e(+i)x = i ex e ix = ex (cos x i cos x + i sin x + sin x) z = + bi Re z =, Im z = b e x sin xdx = ex (sin x cos x) + C e x cos xdx = ex (sin x + cos x) + C
(5) e iθ e iθ = cos(θ + θ ) + i sin(θ + θ ) (cos θ + i sin θ )(cos θ + i sin θ ) = cos θ cos θ sin θ sin θ + i(cos θ sin θ + sin θ cos θ ) cos(θ + θ ) = cos θ cos θ sin θ sin θ sin(θ + θ ) = cos θ sin θ + sin θ cos θ (e iθ ) n = e iθn = cos nθ + i sin nθ = (cos θ + i sin θ) n
(6) P (x) () P (x), Q(x) dx Q(x) Step P (x) = Q (x) OK Step P (x) Q(x) P (x) Q(x) = R(x) + P (x) Q(x) P (x) Q(x) Step3 P (x) Q (x) Q(x) P (x) Q(x). x + 6.7 dx x 3 x + x x + x 3 x + x = x + (x ) x x + (x ) x = A x + B x + C A, B, C (x ) ( ) x + = A(x ) + Bx(x ) + Cx = (A + B)x + ( A B + C)x + A A + B = 0 A B + C = A =, B =, C = 3 A = x + ( ) x 3 x + x = x x + 3 dx (x ) = log x log x 3 x + C = log x x 3 x + C.
(7) ( ) x + b n (3) F x, dx cx + d x + b t = n cx + d x + b cx + d = tn (). 6.8 ( x) dx + x t = + x + x = t, x = t dx, dt = t ( t + ) t tdt = t dt = (t + )(t )dt ( = t = t log t + + C ) t + dt = + x log + x + + C
(8) (4) F (sin x, cos x)dx tn x = t sin x = x x sin cos = x x sin + cos x tn x + tn = t + t cos x, dx dt. sin x = t + t cos x = t + t dx dt = + t 6.9 dx + cos x t = tn x + cos x dx = + t + t dt +t = ( + t ) + t dt = t + 3 dt = t + 3 dt ( ) = tn t + c = tn tn x + C 3 3 3 3
(9) (5) F (e x )dx t = e x dt dx = ex = t () 6.0 dx e x + 4e x + 5 e x = t = t + 4 + 5 t dt t = t + 5t + 4 dt = (t + )(t + 4) dt = ( 3 t + ) dt = t + 4 3 log t + t + 4 + c = ( e x ) 3 log + + C e x + 4 Remrk () (5).,