➜➵✐ ❤ä❝ ➜➭ ◆➼♥❣ ➜➵✐ ❤ä❝ ❙➢ P❤➵♠ ➜➭ ◆➼♥❣ ➜➷♥❣ ❉✉② ❍♦➭♥❣ ❱Ị ✈✐Ư❝ ✈❐♥ ❞ơ♥❣ ♠ét sè ❦✐Õ♥ t❤ø❝ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ➤Ó ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ tr♦♥❣ ❝❤➢➡♥❣ tr×♥❤ tr✉♥❣ ❤ä❝ ♣❤ỉ t❤➠♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝ ◆❣➭♥❤✿ P❤➢➡♥❣ ♣❤➳♣ ❚♦➳♥ s➡ ❝✃♣ ➜➭ ◆➼♥❣✱ ◆➝♠ ✷✵✷✵ ➜➵✐ ❤ä❝ ➜➭ ◆➼♥❣ ➜➵✐ ❤ä❝ ❙➢ P❤➵♠ ➜➭ ◆➼♥❣ ➜➷♥❣ ❉✉② ❍♦➭♥❣ ❱Ị ✈✐Ư❝ ✈❐♥ ❞ơ♥❣ ♠ét sè ❦✐Õ♥ t❤ø❝ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ➤Ó ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ tr♦♥❣ ❝❤➢➡♥❣ tr×♥❤ tr✉♥❣ ❤ä❝ ♣❤ỉ t❤➠♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝ ◆❣➭♥❤✿ P❤➢➡♥❣ ♣❤➳♣ ❚♦➳♥ s➡ ❝✃♣ ▼➲ sè✿ ✽✳✹✻✳✵✶✳✶✸ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ➜➭ ◆➼♥❣✱ ◆➝♠ ✷✵✷✵ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▲ê✐ ❝❛♠ ➤♦❛♥ ✐ ▲ê✐ ❝➯♠ ➡♥ ✐✐ ▼ơ❝ ❧ơ❝ ✐✐✐ ❉❛♥❤ ♠ơ❝ ❦Ý ❤✐Ư✉ ✐✈ ▼ë ➤➬✉ ✈ ❈❤➢➡♥❣ ✶✳ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ➜➵♦ ❤➭♠✱ tÝ♥❤ ❦❤➯ ✈✐✱ tÝ♥❤ ❦❤➯ tÝ❝❤ ❝ñ❛ ❤➭♠ sè ✶ ❜✐Õ♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶ ▼ét sè ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ị ❤➭♠ sè ✶✳✷ ●✐í✐ ❤➵♥ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ❤➭♠ sè ✶ ❜✐Õ♥ sè ✶✳✸ ❈❤➢➡♥❣ ✷✳ ▼ét sè s❛✐ ❧➬♠ ❝đ❛ ❤ä❝ s✐♥❤ tr♦♥❣ ✈✐Ư❝ ✈❐♥ ❞ơ♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝đ❛ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ✶✽ ✷✳✶ ▼ét sè s❛✐ ❧➬♠ tr♦♥❣ ✈✐Ö❝ ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ ❣✐í✐ ❤➵♥ ❝đ❛ ❤➭♠ sè ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ▼ét sè s❛✐ ❧➬♠ tr♦♥❣ ✈✐Ư❝ ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ ✈Ị tÝ♥❤ ❧✐➟♥ tơ❝ ❝ñ❛ ❤➭♠ sè ✳ ✷✷ ✷✳✸ ▼ét sè s❛✐ ❧➬♠ tr♦♥❣ ✈✐Ư❝ ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ ✈Ị tÝ❝❤ ♣❤➞♥ ❝đ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ❈❤➢➡♥❣ ✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱❐♥ ❞ô♥❣ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ sè ➤Ĩ s➳♥❣ t➵♦ ✈➭ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ t♦➳♥ ❤ä❝ tr♦♥❣ ❝❤➢➡♥❣ tr×♥❤ tr✉♥❣ ❤ä❝ ♣❤ỉ t❤➠♥❣ ✸✳✶ ✷✼ ✸✸ ❙➳♥❣ t➵♦ ✈➭ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ ♣❤➢➡♥❣ tr×♥❤✱ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈➭ ❜✃t ♣❤➢➡♥❣ tr×♥❤ ➤➵✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✷ ❙➳♥❣ t➵♦ ✈➭ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ ❣✐í✐ ❤➵♥ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ✸✳✸ ❙➳♥❣ t➵♦ ✈➭ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ✺✹ ❑Õt ❧✉❐♥ ✻✵ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✻✶ ✐✐✐ ❉❛♥❤ ♠ơ❝ ❝➳❝ ❦Ý ❤✐Ư✉ {0, 1, 2, } N : ❚❐♣ ❤ỵ♣ ❝➳❝ sè tù ♥❤✐➟♥✱ ❤❛② N∗ : ❚❐♣ ❤ỵ♣ ❝➳❝ sè tù ♥❤✐➟♥ ❦❤➳❝ ✵✱ ❤❛② R : ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ Z+ : ❚❐♣ ❤ỵ♣ ❝➳❝ sè ♥❣✉②➟♥ [a, b] : ➜♦➵♥ [a, b) : ◆ö❛ ❦❤♦➯♥❣ ∅ : rỗ xX : P tử AB : A [a, b], ❤❛② t❐♣ ❤ỵ♣ [a, b), ❤❛② x ❝đ❛ t❐♣ ❤ỵ♣ t❐♣ ❝♦♥ ❝đ❛ B ✐✈ {1, 2, 3, } {x ∈ R|a ≤ x ≤ b} t❐♣ ❤ỵ♣ X {x ∈ R : a ≤ x < b} ●✐➯✐✳ ❚❛ ❜✐Õ♥ ➤ỉ✐ ✭✸✳✹✵✮ ✈Ị ❞➵♥❣ [x + (x2 + 1)] ex √ f (x) = x2 + √ x + x2 + ex = √ x2 + ′ √ √ = x2 + + x2 + ex ❙❛✉ ➤ã✱ ①Ðt ❤➭♠ sè √ F (x) = x2 + · ex + C ✳ ❚❛ t❤✃② r➺♥❣ √ 2x √ + x2 + ex x2 + (x2 + x + 1) ex √ = x2 + F ′ (x) = = f (x) ❑Õt ❧✉❐♥✿ ❱❐② ❤➭♠ ❱Ý ❞ô ✹✳ F (x) = √ x2 + · ex + C ❧➭ ♥❣✉②➟♥ ❤➭♠ ❝ñ❛ ❤➭♠ sè f (x)✳ ❚×♠ ♥❣✉②➟♥ ❤➭♠ ❝đ❛ ❤➭♠ sè s❛✉ (x − 1)ex f (x) = x2 ●✐➯✐✳ ❚❛ ❜✐Õ♥ ➤ỉ✐ ✭✸✳✹✶✮ ✈Ị ❞➵♥❣ −1 + x2 x f (x) = ❳Ðt ❤➭♠ sè F (x) = ❑Õt ❧✉❐♥✿ ❱❐② ❤➭♠ ·e = ❚❛ t❤✃② r➺♥❣ ′ 1 − 2+ x x F (x) = ex +C x x x ex + C✳ x F (x) = ✸✳✸✳✷ ✭✸✳✹✶✮ ′ + x e x (x − 1) ex e = = f (x) x2 x ❧➭ ♥❣✉②➟♥ ❤➭♠ ❝đ❛ ❤➭♠ sè f (x)✳ ❱❐♥ ❞ơ♥❣ ♠ét sè tÝ♥❤ ❝❤✃t ❜✐Õ♥ ➤ỉ✐ ❝đ❛ ❤➭♠ sè ➤Ĩ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ tÝ❝❤ ♣❤➞♥ ❚Ý♥❤ ❝❤✃t ✶✳ ❈❤♦ f ❧➭ ❤➭♠ sè ❝❤➼♥ ✈➭ ❧✐➟♥ tô❝ tr➟♥ t❛ ❝ã ✈í✐ b ❈❤ø♥❣ ♠✐♥❤✳ ➜➷t x = −t✳ ✈í✐ f (x)dx, −b f (x) dx = ax + b ❑❤✐ ➤ã✱ ✭✸✳✹✷✮ ❑❤✐ ➤ã✱ t❛ ❝ã dx = −dt ✈➭ f (−t) = f (t)✳ t❛ ❝ã b > 0✳ b f (x) dx = ax + −b a > ✈➭ a = 1✳ [−b; b] −f (t) dt = a−t + ✺✻ b b at f (t) dt = at + ax f (x) dx ax + ó ì tế t t ợ b f (x) dx = ax + −b b f (x) dx + ax + f (x) dx ax + −b b b ax f (x) dx + ax + = f (x) dx ax + 0 b f (x)dx = ❱Ý ❞ô ✶✳ ●✐➯✐✳ x4 + x2 + dx✳ 2x + ❚Ý♥❤ tÝ❝❤ ♣❤➞♥ ❚❛ t❤✃② ❤➭♠ sè f −1 ❧➭ ❤➭♠ sè ❝❤➼♥✱ ♥➟♥ ❜➺♥❣ ❝➳❝❤ ➳♣ ❞ô♥❣ tÝ♥❤ ❝❤✃t ✭✸✳✹✷✮✱ t❛ ❝ã x4 + x2 + dx = 2x + −1 x4 + x2 + dx + 2x + x4 + x2 + dx 2x + −1 x4 + x2 + dx = x5 x3 + +x 23 = 15 = ❚Ý♥❤ ❝❤✃t ✷✳ ❈❤♦ f ❧➭ ❤➭♠ sè ❧✐➟♥ tô❝ tr➟♥ ➤♦➵♥ [a, b]✳ ❑❤✐ ➤ã✱ t❛ ❝ã b b f (a + b − x) dx = f (x)dx ✭✸✳✹✸✮ a a ➜➷❝ ❜✐Öt✱ t❛ ❝ã b b f (b − x) dx = ➜➷t t = a + b − x✳ ❑❤✐ ➤ã✱ t❛ ❝ã f (a + b − x) dx = − a a t❛ ♥❤❐♥ ➤➢ỵ❝ ❝➠♥❣ t❤ø❝ b b f (b − x) dx = f (x)dx 0 ✺✼ ❉♦ ➤ã✱ t❛ ❝ã b f (x)dx f (t)dt = f (t)dt = b a = 0✱ dt = −dx✳ b a b ❚r➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❦❤✐ ✭✸✳✹✹✮ 0 ❈❤ø♥❣ ♠✐♥❤✳ f (x)dx a π ❈❤♦ ❱Ý ❞ô ✷✳ π ln b, (a = b)✳ a ln (1 + tan x) dx = I= ❚Ý♥❤ M = a + b✳ ❚❛ t❤✃② ●✐➯✐✳ f (x) = ln(1+tan x) ❧✐➟♥ tô❝ tr➟♥ 0; π ✳ ➳♣ ❞ô♥❣ tÝ♥❤ ❝❤✃t ✭✸✳✹✸✮ ❝❤♦ ✈Ý ❞ơ ♥➭②✱ t❛ ➤➢ỵ❝ π I= π π ln (1 + tan x) dx = π −x ln + tan dx = dx + tan x π π (ln − ln (1 + tan x)) dx = = ln ln 2dx − I π ❚õ ➤➞②✱ t❛ s✉② r❛ 2I = x ln 2|04 ✳ a=8 ❱× t❤Õ✱ t❛ ♥❤❐♥ ➤➢ỵ❝ ❚Ý♥❤ ❝❤✃t ✸✳ ❈❤♦ ❤➭♠ sè ❉♦ ➤ã✱ t❛ ❝ã ✈➭ f b = 2✳ ❙✉② r❛ ❧✐➟♥ tô❝ tr➟♥ π f (cos x))dx ➜➷t [0; 1]✳ ❑❤✐ ➤ã✱ t❛ ❝ã π f (sin x))dx = ❈❤ø♥❣ ♠✐♥❤✳ π ln 2✳ t❛ ❝ã M = a + b = + = 10✳ I= t= π − x✳ ❑❤✐ ➤ã✱ t❛ ❝ã π f (sin x) dx = − f (cos t) dt π π f (cos t) dt = π f (cos x) dx = π ❱Ý ❞ô ✸✳ ❚Ý♥❤ tÝ❝❤ ♣❤➞♥ √ I= √ 2011 2011 sin x2011 √ dx✳ 2011 cos x2011 + sin x2011 ➳♣ ❞ô♥❣ tÝ♥❤ ❝❤✃t ✭✸✳✹✺✮✱ t❛ ❝ã ●✐➯✐✳ π √ 2011 √ 2011 sin x2011 cos x2011 + √ 2011 sin x2011 π √ 2011 dx = ✺✽ √ 2011 cos x2011 √ dx 2011 sin x2011 + sin x2011 ✭✸✳✹✺✮ ◆➟♥ π √ 2011 2I = π = π π sin x2011 cos x2011 + √ 2011 sin x2011 √ 2011 dx + √ sin x2011 √ √ + 2011 2011 cos x2011 + sin x2011 √ 2011 sin x2011 + √ 2011 2011 √ 2011 = √ 2011 cos x2011 √ dx 2011 sin x2011 + sin x2011 √ 2011 √ 2011 cos x2011 √ 2011 sin x2011 + sin x2011 dx √ 2011 cos x2011 √ dx 2011 cos x2011 + sin x2011 π π 1.dx = x|02 = = π ❚õ ➤ã✱ t❛ t❤✉ ➤➢ỵ❝ I= π ✳ ❇➭✐ t❐♣ tù ❧✉②Ư♥ ✶✮ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✳ π I1 = sin x2019 dx cos x2019 + sin x2019 π √ I2 = √ sin x dx √ sin x + cos x ✷✮ f (x)dx = 10✳ ❈❤♦ [2 − 4f (x)] dx✳ ❚Ý♥❤ ❣✐➳ trÞ ❝đ❛ tÝ❝❤ ♣❤➞♥ ✺✾ ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ✈Ị ➤Ị ❱Ị ✈✐Ư❝ ✈❐♥ ❞ơ♥❣ ♠ét sè ❦✐Õ♥ t❤ø❝ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ➤Ó ❣✐➯✐ ❝➳❝ ❜➭✐ t♦➳♥ tr♦♥❣ ❝❤➢➡♥❣ tr×♥❤ tr✉♥❣ ❤ä❝ ♣❤ỉ t❤➠♥❣ t➭✐✿ ✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉ ✶✳ ❍Ö t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ✈Ị ❤➭♠ sè✱ ❣✐í✐ ❤➵♥ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ❤➭♠ sè✱ ➤➵♦ ❤➭♠✱ tÝ♥❤ ❦❤➯ ✈✐✱ tÝ♥❤ ❦❤➯ tÝ❝❤ ❝đ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè✳ ✷✳ ❚r×♥❤ ❜➭② ♠ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ❤➭♠ sè tr♦♥❣ ❣✐➯✐ t♦➳♥ ♣❤ỉ t❤➠♥❣✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❧➭✿ ❝❤Ø r❛ ♠ét sè s❛✐ ❧➬♠ tr♦♥❣ ✈✐Ư❝ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ tÝ♥❤ ❣✐í✐ ❤➵♥✱ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ❤➭♠ sè✱ s❛✐ ❧➬♠ tr♦♥❣ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè✳ ✸✳ ❚r×♥❤ ❜➭② ♠ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ➤➵♦ ❤➭♠ ❤➭♠ sè tr♦♥❣ ❣✐➯✐ t♦➳♥ ♣❤ỉ t❤➠♥❣✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❧➭✿ sư ❞ơ♥❣ ➤➵♦ ❤➭♠ ➤Ĩ ❣✐➯✐ ♠ét sè ❜➭✐ t♦➳♥ ♣❤➢➡♥❣ tr×♥❤✱ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈➭ ❜✃t ♣❤➢➡♥❣ tr×♥❤ ➤➵✐ sè✱ tÝ♥❤ ❣✐í✐ ❤➵♥ ❤➭♠ sè✱ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè✳ ✻✵ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❚➠ ❱➝♥ ❇❛♥ ✭✷✵✶✷✮✱ ●✐➳♦ tr×♥❤ ❣✐➯✐ tÝ❝❤ ✶✱ ◆❤➭ ①✉✃t ❜➯♥ ●✐➳♦ ❞ơ❝✳ ❬✷❪ ➜❐✉ ❚❤Õ ❈✃♣✱ ◆❣✉②Ô♥ ❍✉ú♥❤ P❤➳♥✱ ◆❣✉②Ô♥ ❚❤➳✐ ❙➡♥✱ ❚r➬♥ ➜×♥❤ ❚❤❛♥❤ ✭✷✵✵✼✮✱ ●✐➯✐ tÝ❝❤ ❚♦➳♥ ❤ä❝✱ ◆❤➭ ①✉✃t ❜➯♥ ●✐➳♦ ❞ơ❝✳ ❬✸❪ ◆❣✉②Ơ♥ ❚➭✐ ❈❤✉♥❣ ✭✷✵✶✺✮✱ ❙➳♥❣ t➵♦ ✈➭ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤✱ ❤Ư ♣❤➢➡♥❣ tr×♥❤ ✈➭ ❜✃t ♣❤➢➡♥❣ trì t ổ ợ ố í ▼✐♥❤✳ ❬✹❪ P❤➵♠ ❍å♥❣ ❉❛♥❤ ✭✷✵✵✼✮✱ ●✐➳♦ tr×♥❤ t♦➳♥ ❝❛♦ ❝✃♣ ❣✐➯✐ tÝ❝❤✱ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ t ố í ứ ỗ ❍♦➭♥❣ ❍➭✱ ▲➟ ❍♦➭♥❣ ◆❛♠✱ ➜♦➭♥ ▼✐♥❤ ❈❤➞✉✱ ➜➭♦ ❚❤Þ ◆❣ä❝ ❍➭✱ P❤➢➡♥❣ ♣❤➳♣ ❣✐➯✐ ❝➳❝ ❞➵♥❣ t♦➳♥ ❚❍P❚ ✭◆❣✉②➟♥ ❤➭♠✱ ❚Ý❝❤ ♣❤➞♥ ✈➭ ø♥❣ ❞ô♥❣✮✱ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐✳ ❬✻❪ ◆❣✉②Ô♥ ❱➝♥ ▲é❝✱ ◆❣✉②Ô♥ ❱✐Õt ➜➠♥❣ ✭✷✵✶✵✮✱ ❈❤✉②➟♥ ➤Ò ➤➵♦ ❤➭♠ ✈➭ ❦❤➯♦ s➳t ❤➭♠ sè ✭❇å✐ ❞➢ì♥❣ ❤ä❝ s✐♥❤ ❣✐á✐✲❧✉②Ư♥ t❤✐ ➤➵✐ ❤ä❝✮✱ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ t❤➭♥❤ ♣❤è ❍å ❈❤Ý ▼✐♥❤✳ ❬✼❪ ◆❣✉②Ơ♥ ❱➝♥ ▼❐✉✱ ◆❣✉②Ơ♥ ❚❤đ② ❚❤❛♥❤ ✭✷✵✵✹✮✱ ❈❤✉②➟♥ ➤Ị ❜å✐ ❞➢ì♥❣ ❤ä❝ s✐♥❤ ❣✐á✐ t♦➳♥ ❚❍P❚ ✭●✐í✐ ❤➵♥ ❞➲② sè ✈➭ ❤➭♠ sè✮✱ ◆❤➭ ①✉✃t ❜➯♥ ●✐➳♦ ❞ô❝✳ ❬✽❪ ❈❤✉ ❚rä♥❣ ❚❤❛♥❤✱ ❚r➬♥ ❚r✉♥❣ ✭✷✵✶✶✮✱ ❈➡ së t♦➳♥ ❤ä❝ ❤✐Ư♥ ➤➵✐ ❝đ❛ ❦✐Õ♥ t❤ø❝ ♠➠♥ t♦➳♥ ♣❤ỉ t❤➠♥❣✱ ◆❤➭ ①✉✃t ❜➯♥ ❣✐➳♦ ❞ô❝✳ ❬✾❪ ❚r➬♥ ❉✉② ❚❤ø❝ ✭✷✵✶✼✮✱ ▲➭♠ ❝❤đ ❜➭✐ t❐♣ tr➽❝ ♥❣❤✐Ư♠ ❤➭♠ sè ✈➭ ø♥❣ ❞ơ♥❣ ✭↕♥ ❧✉②Ư♥ t❤✐ ❚❍P❚◗●✱ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❝❤♦ ❣✐➳♦ ✈✐➟♥✮✱ ◆❤➭ ①✉✃t ❜➯♥ ❍å♥❣ ➜ø❝✳ ❬✶✵❪ ▲➟ ❱➝♥ ❚rù❝ ✭✷✵✵✵✮✱ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝✱ t❐♣ ✶✱ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐✳ ✻✶ ... số số ợ số ợ ột sè ❤➭♠ sè s➡ ❝✃♣✳ ✶✳✶✳✶ R✱ ❬❬✶✵❪❀ ❚r❛♥❣ ✶✻❪ ❈❤♦ ➜Þ♥❤ ♥❣❤Ü❛✳ ➳♥❤ ①➵ f : X → Y X, Y ❧➭ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ t❐♣ ❝➳❝ sè t❤ù❝ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❤➭♠ sè ♠ét ❜✐Õ♥ sè tự t X ợ ọ t ị ủ số. .. tr×♥❤ ❝❤ø❛ ❝➝♥ t❤ø❝✳ ◆é✐ ❞✉♥❣ ❝đ❛ ♣❤➢➡♥❣ ♣❤➳♣ ❧➭ ✈✐Ư❝ t❤➟♠ ❜ít ột số ệ số ị trì ❝❤♦✱ s❛✉ ➤ã tõ ❣✐➯ t❤✐Õt ➤➲ ❝❤♦ t❛ sÏ tì r ợ ệ số ụ tể ế ổ tr♦♥❣ ❜➭✐ t♦➳♥✳ ❈❤ó♥❣ t➠✐ sÏ tr×♥❤ ❜➭② ❦Ü t❤✉❐t ♥➭②... ✭♥❣❤Ü❛ ❧➭ ♥Õ✉ tö y∈Y y∈Y ♠ét ❤➭♠ sè tõ t❐♣ f ❧➟♥ t❐♣ x , x2 ∈ X Y Y✳ ✈➭ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❧➭ ❤➭♠ số ợ t ó ộ tị số X x1 = x2 x X tì ỗ ❑❤✐ ➤ã✱ t❛ ❝ã t❤Ó ➤➷t t➢➡♥❣ ♥❣❤Ü❛ ❧➭ x = f −1 (y) y → x = f −1