ëY5 È©Ò È©Òe¦ È©¹ëê §13.2 ¹ëCþ~ÂÈ© 13.2.1 ¹ëCþ~ÂÈ©9Ù5 ¼ê f(x, u) 3«m [a, b] × [α, β] þëY, éu?½ u ∈ [α, β], ¼ê f(x, u) éCþ x 3 [a, b] þ Riemann È, ù¡È© Z b a f(x, u)dx ´¹ëCþ u ~ÂÈ©. §½Â ¼ê u 7−→ ϕ(u) = Z b a f(x, u)dx. ù!Ì8, Ò´?عëCþ~ÂÈ©5. 1/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê ½n 1 XJ¼ê f(x, u) 3 I = [a, b] × [α, β] þëY, K ϕ(u) = Z b a f(x, u)dx 3 [α, β] þëY. y² 3«m [α, β] þ?: u0, u´ |ϕ(u) − ϕ(u0)| = Z b a f(x, u)dx − Z b a f(x, u0)dx 6 Z b a |f(x, u) − f(x, u0)|dx, du f(x, u) 34« I þëY, 7ëY. é?¿ ε > 0, 3ê δ, I ¥ü: (x1, u1) (x2, u2) ålu δ, Òk |f(x1, u1) − f(x2, u2)| < ε. 2/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê AO |u − u0| < δ , é?¿ x ∈ [a, b] Ñk |f(x, u) − f(x, u0)| < ε, l |ϕ(u) − ϕ(u0)| < (b − a)ε. ùÒy² ϕ(u) 3: u0 ?ëY, d u0 ?¿5, ϕ(u) 3 [α, β] þë Y. du lim u→u0 ϕ(u) = ϕ(u0) ¤ lim u→u0 Z b a f(x, u)dx = Z b a lim u→u0 f(x, u)dx, Ò´`4$È©$gS±. 3/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê ~ 1 f(x) ´«m [0, 1] þëY¼ê, ?ؼê F(t) = Z 1 0 tf(x) x2 + t 2 dx ëY5. ) éz½ t ∈ R, ¼ê h(x, t) = tf(x) x2 + t 2 Ñ´'u x ëY¼ê, Ïd, F(t) ´½Âмê, ´Û¼ê. 0 < α < β. Ï h(x, t) 3 [0, 1] × [α, β] þëY, ¤± F(t) 3 [α, β] ëY, l F(t) 3 t 6= 0 ?Ñ´ëY. éu 0 < t < 1, k Z 1 0 tf(x) x2 + t 2 dx = Z t 1/3 0 tf(x) x2 + t 2 dx + Z 1 t 1/3 tf(x) x2 + t 2 dx. Ï f 3 [0, 1] þëY, |f(x)| 6 M. Ï Z 1 t 1/3 tf(x) x2 + t 2 dx 6 t t 2/3 + t 2 M = t 1/3 1 + t 4/3 M → 0, (t → 0 + ). 4/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê â1È©¥½ní2, 3 ξ ∈ (0, t1/3 ) ¦ Z t 1/3 0 tf(x) x2 + t 2 dx = f(ξ) Z t 1/3 0 t x2 + t 2 dx = f(ξ) arctan x t x=t 1/3 x=0 = f(ξ) arctan 1 t 2/3 . Ï lim t→0+ Z t 1/3 0 tf(x) x2 + t 2 dx = π 2 f(0). u´ lim t→0+ F(t) = π 2 f(0). Ón lim t→0− F(t) = − π 2 f(0). dd f(0) = 0 , F(t) 3 t = 0 ëY, f(0) 6= 0 , F(t) 3 t = 0 ØëY. 5/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê 3(½ ϕ(u) ´ u ëY¼ê, ÒkU5 §3«m [α, β] þÈ© Z β α ϕ(u)du = Z β α Z b a f(x, u)dx du. ¼ê f(x, u) 3 I þëY, þªmàÈ© u f(x, u) 3 I þÈ ©, ¤ Z β α Z b a f(x, u)dx du = Z b a Z β α f(x, u)du dx. ùB´ ½n 2 XJ¼ê f(x, u) 3 I = [a, b] × [α, β] þëY, K ϕ(u) = Z b a f(x, u)dx 3 [α, β] È, ¿k Z β α ϕ(u)du = Z β α Z b a f(x, u)dx du = Z b a Z β α f(x, u)du dx. 6/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê ~ 2 O Z 1 0 x b − x a ln x dx, 0 < a < b. ) 5¿ x b − x a ln x = Z b a x u du, ¼ê h(x, u) = x u 3 [0, 1] × [a, b] ëY, Ï k I = Z 1 0 Z b a x udu dx = Z b a Z 1 0 x u dx du = Z b a 1 u + 1 x u+1 1 0 du = Z b a 1 u + 1 du = ln b + 1 a + 1 . 7/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê y3?Úïļê ϕ(u) 5. ½n 3 ¼ê f(x, u) 3 I = [a, b] × [α, β] þëY, é u këY ê, K¼ê ϕ(u) = Z b a f(x, u)dx 3 [α, β] þ, ¿k ϕ 0 (u) = Z b a ∂f(x, u) ∂u dx. y² - Z b a ∂f(x, u) ∂u dx = g(u), K g(u) ´ [α, β] þëY¼ê, â½n 2, α 6 v 6 β k 8/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê Z v α g(u)du = Z b a Z v α ∂f(x, u) ∂u du dx = Z b a [f(x, v) − f(x, α)]dx = ϕ(v) − ϕ(α). d½n 1 , g(u) ´ [α, β] þëY¼ê, ϕ(v) ´ g(v) ¼ê, Ïd ϕ 0 (v) = g(v). ùÒ´¤y²úª. 9/20 kJ Ik J I £ ¶ '4 òÑ
ëY5 È©Ò È©Òe¦ È©¹ëê ~ 3 Á¦È© Z 1 0 ln(1 + x) 1 + x2 dx . ) ĹëCþÈ© I(u) = Z 1 0 ln(1 + ux) 1 + x2 dx. ùȩȼê ln(1+ux) 1+x2 9Ù'u u û x (1+x2)(1+ux) Ñ3 [0, 1]2 þ ëY, d½n 3 Òk I 0 (u) = Z 1 0 x (1 + x2)(1 + ux) dx = 1 1 + u2 Z 1 0 x 1 + x2 + u 1 + x2 − u 1 + ux dx = 1 1 + u2 ln 2 2 + π 4 u − ln(1 + u) . òdªüà'u u l 0 1 È© 10/20 kJ Ik J I £ ¶ '4 òÑ