MA TRẬN ĐỊNH THỨC HỆ PHƯƠNG TRÌNH TUYẾN TÍNH

Ch÷ìng 4 MA TR�N �ÀNH THÙC H� PH×ÌNG TR�NH TUY�N T�NH 4 1 Ma trªn 4 1 1 Kh¡i ni»m ma trªn ? �ành ngh¾a 4 1 Mët b£ng sè chú nhªt câ m h ng n cët �÷ñc gåi l mët ma trªn cï m×n aij l ph¦n tû n¬m ð h ng i. taài liệu cao đẳng đại học, tài liệu luận văn, giáo trình thạc sy, tiến sỹ, tài liệu THCS MA TRẬN ĐỊNH THỨC HỆ PHƯƠNG TRÌNH TUYẾN TÍNH

ữỡ Pì ❚❘➐◆❍ ❚❯❨➌◆ ❚➑◆❍ ✹✳✶✳ ▼❛ tr➟♥ ✹✳✶✳✶✳ ❑❤→✐ ♥✐➺♠ ♠❛ tr➟♥ ✣à♥❤ ♥❣❤➽❛ ✹✳✶✳ aij ▼ët ❜↔♥❣ sè ❝❤ú ♥❤➟t ❝â ❧➔ ♣❤➛♥ tû ♥➡♠ ð ❤➔♥❣ ❺ A= i a11 a12 a21 a22 ··· ··· am1 am2 ❉↕♥❣ t❤✉ ❣å♥ ❧➔ ❝ët A = [aij ]m×n ❱➼ ❞ư ✹✳✶✳ m ❤➔♥❣ n ❝ët ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠❛ tr➟♥ ❝ï m × n✳ A✳ ❚❛ ✈✐➳t✿ ❝õ❛ ♠❛ tr➟♥ · · · a1n · · · a2n ··· ··· à à à amn ỵ t tr ù ã j mìn a11 a12 a21 a22 A= ··· ··· am1 am2 ❤♦➦❝  · · · a1n · · · a2n   ··· ···  · · · amn A = (aij )m×n ✳ ❧➔ ❇↔♥❣ sè Mm×n ➉ ➌ −2 A= −7 ❧➔ ♠ët ♠❛ tr➟♥ ❝ï 2×3 ✈ỵ✐ ❝→❝ ♣❤➛♥ tû a11 = 1; a12 = −2; a13 = 4; a21 = 3; a22 = 5; a23 = ú ỵ r ổ ♥➔②✱ ❝❤ó♥❣ t❛ ❝❤➾ ①➨t ❝❤õ ②➳✉ ❝→❝ ♠❛ tr➟♥ t❤ü❝✱ tù❝ ❧➔ ❝→❝ ♠❛ tr➟♥ ✈ỵ✐ aij ∈ R ✹✳✶✳✷✳ ▼ët sè ❞↕♥❣ ✤➦❝ ❜✐➺t ❝õ❛ ♠❛ tr➟♥ ❛✮ ▼❛ tr➟♥ ❦❤æ♥❣ ▼❛ tr➟♥ ❦❤æ♥❣ ❧➔ ♠❛ tr➟♥ ♠➔ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ✤➲✉ ❜➡♥❣ ❦❤æ♥❣✳ ▼❛ tr➟♥ ổ ữủ ỵ O ❚❍Ù❈ ✲ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❚❯❨➌◆ ❚➑◆❍ ✽✽ ❱➼ ❞ư ✹✳✷✳ • ➉ ❧➔ ♠ët ♠❛ tr➟♥ ❦❤ỉ♥❣ ❝ï ❜✮ ➌ 0 0 0 0 2×4 ▼❛ tr➟♥ ❤➔♥❣✱ ♠❛ tr➟♥ ❝ët ▼❛ tr➟♥ ❤➔♥❣ ❧➔ ♠❛ tr➟♥ ❝❤➾ ❝â ♠ët ❤➔♥❣✳ ▼❛ tr➟♥ ❝ët ❧➔ ♠❛ tr➟♥ ❝❤➾ ❝â ♠ët ❝ët✳ ❝✮   2 ❧➔ ❧➔ ♠❛ tr➟♥ ❤➔♥❣; ♠❛ tr➟♥ ❝ët ▼❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n n ❤➔♥❣ ✈➔  a11 a12  a21 a22  A= · · · · · · an1 an2 ▼❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ ♥ ❧➔ ♠❛ tr➟♥ ❝â ❈→❝ ♣❤➛♥ tû a11 , a22 , , ann ✤÷đ❝ ❣å✐ n ởt ỵ A = [aij ]n ❤♦➦❝ A = (aij )n  · · · a1n · · · a2n    ✳✳ ✳ · · · · · · ann ♣❤➛♥ tû ❝❤➨♦✱ ❝❤ó♥❣ t t ữớ tr ổ ỵ ❤✐➺✉ t➟♣ ❝→❝ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ❧➔ Mn ❞✮ ▼❛ tr➟♥ t❛♠ ❣✐→❝✿ ▼❛ tr➟♥ ✈✉æ♥❣ A = [aij ]n tr♦♥❣ ✤â aij = ♥➳✉ i>j ✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥✳ ✭▼❛ tr➟♥ ❝â tỷ ữợ ữớ ❜➡♥❣ ✵✮  a11 a12 · · ·  a22 · · ·   · · · · · · ✳ ✳ ✳ 0 ··· ▼❛ tr➟♥ ✈✉æ♥❣ A = [aij ]n  a1n a2n    · · · ann tr♦♥❣ ✤â   a11 a12 · · · a1n  a22 · · · a2n      ✳✳  ✳ · · · ann ❝á♥ ✈✐➳t aij = ♥➳✉ i ❧➔ ❤➡♥❣ sè ❜➜t ❦ý✳ √ dx ✳ ❚❛ ❝â x2 +a √ √ dx ,a > a2 −x2 a sin t, − π2 < 0✳ = √ (x+ x2 +a)dx √ ✱ ❞♦ ✤â x2 +a √ √ dx a2 −x2 dx = = ln |x + √ x2 + a| + C −a < x < a✳ a cos tdt a cos t = π = = t + C ⇒ t = arcsin xa dx ✳ a2 +x2 adt , a2 cos2 t = a x✳ ✣➦t x = a tan t, − π2 < t < + x2 = a2 + a2 tan2 t = a2 (1 + tan2 t) = = adt cos2 t a2 /cos2 t x = a tan t, − π2 < t < dx a2 +x2 √ dx a2 −x2 dt = t + C ⇒ = arcsin xa + C dx a2 +x2 ❱➟② √ dx x2 +a x2 + a| + C ⇒ ❍➔♠ ữợ t ợ õ ln | x−a | + C, x+a t < π2 ✳ ❉➵ t❤➜② ♣❤➨♣ ✤ê✐ ❜✐➳♥ ♥➔② t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✐✮ ❝õ❛ ✤à♥❤ √ √ ❝â dx = a cos tdt, a2 − x2 = a2 − a2 sin2 t = a2 cos2 t = a| cos t| = a cos t ✈➻ − π2 < t < π2 ✳ ❙✉② r❛ ✭❞✮ ❚➼♥❤ ❚❛ ❝â 2a ln | x−a | + C1 − C2 x+a ❚➟♣ ①→❝ ữợ t x = a sin t, − π2 < t < ❱➟② 2a ln |x + a| − C2 = √ (x+ x2 +a)dx √1 √ ✳ x+ x2 +a x2 +a = = ln |t| + C = ln |x + √ dx a2 −x2 ❚❛ ❝â √ dx x2 +a x2 + a ⇒ dt = dt t = 2a ❙✉② r❛ π = dt a π ✳ a2 ✳ ❙✉② r❛ cos2 t = a1 t + C ⇒ dx a2 +x2 = a1 t + C ⇔ t = arctan xa arctan xa + C ❆✳✸✳✶✳✷✳✷✳ P❤÷ì♥❣ ♣❤→♣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ●✐↔ sû u, v ❧➔ ❤❛✐ ❤➔♠ sè ❝â (a, b) ✈➔ t❤ä❛ ♠➣♥ ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣ (a, b)✳ ❑❤✐ ✤â u.v ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (u(x)v(x)) = u (x)v(x) + u(x)v (x) x ∈ (a, b)✳ ❈→❝ sè ❤↕♥❣ ð ✈➳ ♣❤↔✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ✤➲✉ ❧✐➯♥ tö❝ ♥➯♥ ❦❤♦↔♥❣ (a, b)✳ ❉♦ ✤â t❤❡♦ t➼♥❤ ❝❤➜t t❤ù ❤❛✐ ❝õ❛ t➼❝❤ ♣❤➙♥ ❜➜t ✤à♥❤ t❛ ❝â ✈ỵ✐ ♠å✐ tr➯♥ (u(x)v(x)) dx = u (x)v(x)dx + ❚➼❝❤ ♣❤➙♥ ð ✈➳ tr→✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ❜➡♥❣ u(x)v(x) ❝â ♥❣✉②➯♥ ❤➔♠ u(x)v (x)dx ✈➻ ✤↕♦ ❤➔♠ ❝õ❛ u(x)v(x) số ữợ t số t tr t ỗ s t ♣❤➙♥ ð ✈➳ ♣❤↔✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥✮✳ ❚ø ✤â t❤❛② u(x)v(x) = u (x)dx = du(x), v (x)dx = dv(x) v(x)du(x) + u(x)dv(x) ⇒ t❛ ✤÷đ❝ u(x)dv(x) = u(x)v(x) − v(x)du(x) ❆✳✸ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ✶✹✼ ❱➟② t❛ ❝â ❝æ♥❣ t❤ù❝ s❛✉✱ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ • u(x)dv(x) = u(x)v(x) − v(x)du(x) v du = ❱➼ ❞ö ❆✳✶✷✳ √ ✭❛✮ ❚➼♥❤ ✣➦t u= √ x2 + adx✳ x2 + a ✈➔ ❝❤å♥ ❝â s❛♦ ❝❤♦ dv = dx✳ ❑❤✐ ✤â √xdx ✈➔ ❝â t❤➸ ❝❤å♥ x2 +a v = x✳ ❚❛ √ √ x2 + adx = x x2 + a − x √xdx x2 +a √ √ adx 2 √ =x x +a− x + adx + x2 +a ⇒2 √ √ x2 + adx = x x2 + a + √adx x2 +a ⇒ √ x2 + adx = x √ x2 + a + a √ dx x2 +a ❚ø ✤â →♣ ❞ö♥❣ ❦➳t q✉↔ ❝õ❛ ✈➼ ❞ö ❆✳✶✶ ♣❤➛♥ ✭❜✮ t❛ ✤÷đ❝ √ x2 + adx = x √ √ x2 + a + a2 ln |x + x2 + a| + C √ ✭❜✮ ❚➼♥❤ ✣➦t ❚❛ ❝â ⇒2 u= √ a2 − x2 dx, a > 0✳ a2 − x2 ✈➔ ❝❤å♥ v s❛♦ ❝❤♦ dv = dx✳ ❑❤✐ ✤â du = √−xdx ✈➔ ❝â t❤➸ ❝❤å♥ a2 −x2 v = x✳ √ √ a2 − x2 dx = x a2 − x2 − x √−xdx a2 −x2 √ √ = x a2 − x − a2 − x2 dx + √aa2dx −x2 √ √ a2 − x2 dx = x a2 − x2 + √a dx a2 −x2 ⇒ √ a2 − x2 dx = x √ a2 − x + a2 √ dx a2 −x2 ❚ø ✤â →♣ ❞ö♥❣ ❦➳t q✉↔ ❝õ❛ ✈➼ ❞ö ❆✳✶✶ ♣❤➛♥ ✭❝✮ t❛ ✤÷đ❝ √ a2 − x2 dx = x √ a2 − x + a2 arcsin xa + C ✶✳✸✳✶✳✷✳✸✳ ❇↔♥❣ t➼❝❤ ♣❤➙♥ ❝ì ❜↔♥ ❚ø ❜↔♥❣ ✤↕♦ ❤➔♠ ❝→❝ ❤➔♠ sè ❝➜♣ ❝ì ❜↔♥ ✭♠ư❝ ✶✳✷✳✶✳✷✳✸✮ ✈➔ ❝→❝ ❦➳t q✉↔ ❝õ❛ ♠ư❝ ✶✳✸✳✶✳✷ t❛ ♥❤➟♥ ✤÷đ❝ ❜↔♥❣ t➼❝❤ ♣❤➙♥ ❝→❝ ❤➔♠ t❤ỉ♥❣ ❞ư♥❣ s❛✉✳ 0dx = C; α+1 xα dx = xα+1 + C, α = −1❀ x ax dx = lna a + C ❀ sin xdx = − cos x + C ❀ dx = − cot x + C ❀ sin2 x dx √ = arcsin xa + C ❀ a2 −x2 dx = a1 arctan√xa + C ❀ a2 +x2 √ dx = ln |x + x2 + a| + C ❀ x2 +a ❆✳✸✳✷✳ ❚➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ dx = x + C ❀ dx = ln |x| + C ❀ x x e dx = ex + C ❀ cos xdx = sin x + C ❀ dx = tan x + C ❀ cos2 x √ √ 2 a − x2 dx = x2 a2 − x2 + a2 arcsin xa + C ❀ dx = 2a ln | x−a x2 −a2 x+a √ √ √ | + C❀ a x x + adx = x2 + a + ln |x + x2 + a| + C ✳ P❍➆P ❚➑◆❍ ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ❙➮ ✶✹✽ ❆✳✸✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ❚r♦♥❣ ♠ö❝ ♥➔② t❛ ❝➛♥ ❦❤→✐ ♥✐➺♠ ✈➲ tr ú ữợ ú ởt t sè t❤ü❝✳ ❈❤♦ E ❧➔ t➟♣ sè t❤ü❝ ❦❤→❝ ré♥❣✳ tỗ t số tr số y y s❛♦ ❝❤♦ ✤÷đ❝ ❣å✐ ❧➔ ❝➟♥ tr➯♥ ❝õ❛ t➟♣ ữợ ữủ tữỡ tỹ t E x ≤ y, ∀x ∈ E t❤➻ t➟♣ E ✤÷đ❝ E tr ữợ t❤➻ E ✤÷đ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥✳ ✣à♥❤ ♥❣❤➽❛ ❆✳✶✻✳ ◆➳✉ t➟♣ ❣å✐ ❧➔ ❝➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ E E ❜à ❝❤➦♥ tr➯♥ t❤➻ ❝➟♥ tr➯♥ ♥❤ä ♥❤➜t tr♦♥❣ sè tr ữủ ữủ ỵ sup E sup {x} t E ữợ t xE ữợ ợ t tr số ữợ ữủ ữợ ú inf E E ữủ ỵ inf {x} xE ❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ ❞➵ t❤➜② ♠é✐ t➟♣ sè t❤ü❝ ❝â ❦❤ỉ♥❣ q✉→ ♠ët ❝➟♥ tr➯♥ ✤ó♥❣✱ ✈➔ ❦❤ỉ♥❣ q✉→ ởt ữợ ú ỹ tỗ t tr ú ữủ ự tr ỵ ỹ tỗ t ữợ ú ữủ ự tữỡ tü✳ ✣à♥❤ ♥❣❤➽❛ ❆✳✶✼✳ [a, b]✱ ❛ ❁ ❜✳ ❈❤✐❛ ✤♦↕♥ [a, b] t❤➔♥❤ ♥ ✤♦↕♥ ♥❤ä ❜ð✐ ❝→❝ ✤✐➸♠ x0 , x1 , , xn : x0 = a < x1 < < xn = b✳ ❚➟♣ P = {x0 , x1 , , xn } ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤➙♥ ✤✐➸♠ ❝õ❛ ✤♦↕♥ [a, b]✳ ✣➦t ●✐↔ sû ❤➔♠ f ①→❝ ✤à♥❤ ✈➔ ❜à ❝❤➦♥ tr➯♥ ✤♦↕♥ ∆xi = xi − xi−1 , i = 1, n, Mi = sup {f (x)|x ∈ [xi−1 , xi ]}, i = 1, n, mi = inf {f (x)|x ∈ [xi−1 , xi ]}, i = 1, n, n U (P) = Mi ∆xi , i=1 n L(P) = mi ∆xi , i=1 U = inf U (P) ✭❆✳✷✻✮ L = sup L(P) ✭❆✳✷✼✮ tr♦♥❣ ✤â ❝➟♥ tr➯♥ ✤ó♥❣ ✈➔ ữợ ú t tt ❝õ❛ ✤♦↕♥ ② y = f (x) ① ❖ ❛ x0 ξ1 x1 ξ2 x2 ξi ❜ xn ❍➻♥❤ ❆✳✹ [a, b]✳ ❆✳✸ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ◆➳✉ U =L t❤➻ t❛ ♥â✐ ❤➔♠ f ✶✹✾ [a, b]✱ ❣✐→ trà ❝❤✉♥❣ ❝õ❛ ❝→❝ ✤↕✐ ❧÷đ♥❣ ✭❆✳✷✻✮ ✈➔ f (x)dx t tr s ữủ ỵ ❤✐➺✉ ❧➔ b a ✈➔ ✤÷đ❝ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ❝õ❛ ❤➔♠ f tr➯♥ ✤♦↕♥ [a, b]✳ ❈→❝ ỵ a, b, x, f (x), f (x)dx ữủt ữủ ữợ tr t số t tự ữợ ❞➜✉ t➼❝❤ ♣❤➙♥✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ t❤➯♠✿ ◆➳✉ ✈➔ ❤➔♠ a>b f ①→❝ ❱➻ ❤➔♠ f f ✈➔ ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ ❬❜✱ ❛❪✱ t❤➻ a ✤à♥❤ t↕✐ ❛✱ t f (x)dx = a tỗ t↕✐ ❝→❝ sè m ✈➔ M b a f (x)dx = − a b f (x)dx, ❝á♥ ♥➳✉ a=b s❛♦ ❝❤♦ m ≤ f (x) ≤ M, ∀x ∈ [a, b] ◆❣❤➽❛ ❧➔ ✈ỵ✐ ♠å✐ ♣❤➙♥ ✤✐➸♠ P m(b − a) ≤ L(P) ≤ U (P) ≤ M (b − a), ✈➻ ✈➟② ❝→❝ sè U (P) ✈➔ L(P) t↕♦ t❤➔♥❤ ❝→❝ t➟♣ ❜à ❝❤➦♥✳ ✣✐➲✉ ✤â ❝❤ù♥❣ tä r➡♥❣ ữủ tỗ t ✈➲ sü trị♥❣ ♥❤❛✉ ❣✐ú❛ ❝→❝ ✤↕✐ ❧÷đ♥❣ ➜②✱ ❤❛② t➼♥❤ ❦❤↔ t➼❝❤ ❝õ❛ ❤➔♠ f tr➯♥ ✤♦↕♥ [a, b]✱ ❧➔ ✈➜♥ ✤➲ t✐♥❤ t➳ ❤ì♥✳ ❚❛ s➩ ①❡♠ ①➨t ✈➜♥ ✤➲ ➜② tr♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦✳ ❆✳✸✳✷✳✷✳ ✣✐➲✉ ❦✐➺♥ ❦❤↔ t➼❝❤ ❚❛ ❣å✐ ♣❤➙♥ ✤✐➸♠ ❝❤ù❛ ♥❤✐➲✉ ❤ì♥ ✈➔ xi P Q ❝õ❛ ✤♦↕♥ [a, b] P ♥➳✉ Q P rữợ t sỷ Q ❧➔ x ✳ ●✐↔ sû xi−1 < x < xi ✈ỵ✐ xi−1 ❧➔ sü t→♥ ♥❤ä ❝õ❛ ✤ó♥❣ ♠ët ✤✐➸♠✳ ỵ ợ t ❝õ❛ ♣❤➙♥ ✤✐➸♠ P✳ ✣➦t ω1 = inf{f (x)|x ∈ [xi−1 , x∗ ]}, ω2 = inf{f (x)|x ∈ [x∗ , xi ]} ❘ã r➔♥❣ ❧➔ ω1 ≥ mi ✈➔ mi tr õ ữ trữợ mi = inf {f (x)|x ∈ [xi−1 , xi ]} ❚ù❝ ❧➔ L(Q) − L(P) = ω1 (x∗ − xi−1 ) + ω2 (xi − x∗ ) − mi (xi − xi−1 ) = (ω1 − mi )(x∗ − xi−1 ) + (ω2 − mi )(xi − x∗ ) ≥ 0, ❤❛② L(Q) ≥ L(P )✳ ◆➳✉ Q ❝❤ù❛ ♥❤✐➲✉ ❤ì♥ P k ✤✐➸♠ t❤➻ ❧➦♣ ❧↕✐ ❧➟♣ ❧✉➟♥ tr➯♥ k ❧➛♥ t❛ ♥❤➟♥ ✤÷đ❝ L(P) ≤ L(Q) ✭❆✳✷✾✮ U (Q) ≤ U (P) ✭❆✳✸✵✮ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ✤÷đ❝ ỵ ố U tr ổ tự ✈➔ sè L tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭❆✳✷✼✮ t❤ä❛ ♠➣♥ L ≤ U P❍➆P ❚➑◆❍ ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ❙➮ ✶✺✵ ∆✳ ●✐↔ sû P1 ✈➔ P2 [a, b]✱ P = P1 ∪ P2 ✳ ❧➔ ❤❛✐ ♣❤➙♥ ✤✐➸♠ ❝õ❛ ❚❤❡♦ ✭❆✳✷✾✮ ✈➔ ✭❆✳✸✵✮ L(P1 ) ≤ L(P) ≤ U (P) ≤ U (P2 ) ❚ù❝ ❧➔ L(P1 ) ≤ U (P2 ) ❈♦✐ P2 ❝è ✤à♥❤✱ ❧➜② ❝➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ t➟♣ ✭❆✳✸✶✮ {L(P1 )} t❤❡♦ t➜t ❝↔ ❝→❝ ♣❤➙♥ ✤✐➸♠ P1 ✱ tø ✭❆✳✸✶✮ t❛ ữủ L U (P2 ) ữợ ú t ỵ {U (P2 )} t❤❡♦ t➜t ❝↔ P2 ✭❆✳✸✷✮ tr♦♥❣ ✭❆✳✸✷✮ t❛ ♥❤➟♥ ✤÷đ❝ ỵ f t tr➯♥ ✤♦↕♥ [a, b] ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ > tỗ t P s ∆✳ U (P) − L(P) < ε ❱ỵ✐ ♠å✐ ♣❤➙♥ ✤✐➸♠ P ✭❆✳✸✸✮ t❛ ❝â L(P) ≤ L ≤ U ≤ U (P) ❱➻ ✈➟② tø ✭❆✳✸✸✮ s✉② r❛ ∀ε > ❝â ❜➜t ✤➥♥❣ t❤ù❝✿ ≤ U − L < ε ◆❣❤➽❛ ❧➔✱ ✈ỵ✐ ♠å✐ tr➯♥ ✤♦↕♥ P2 ❜➜t ✤➥♥❣ t❤ù❝ ✭❆✳✸✹✮ ✤÷đ❝ t❤ä❛ ♠➣♥✱ ❞♦ ✤â U = L✱ ❤❛② f ❦❤↔ t➼❝❤ [a, b]✳ ❇➙② ❣✐í ❣✐↔ sû ✈➔ ε>0 ✭❆✳✸✹✮ f ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b] ✈➔ ❝❤♦ sè ε > 0✳ ❑❤✐ ✤â tỗ t P1 tọ b f (x)dx < ε ✭❆✳✸✺✮ f (x)dx − L(P1 ) < ε ✭❆✳✸✻✮ U (P2 ) − a b a ❈❤å♥ P = P1 ∪ P2 ✳ ❑❤✐ ✤â ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭❆✳✷✾✮✱ ✭❆✳✸✵✮✱ ✭❆✳✸✺✮✱ ✭❆✳✸✻✮ ❝❤♦ t❤➜② b U (P) ≤ U (P2 ) < f (x)dx + ε < L(P1 ) + ε ≤ L(P) + , a ố ợ ữủ tọ ỷ ỵ ữớ t ự ữủ t t ợ q trồ ỵ f tử tr➯♥ [a, b] t❤➻ f ❦❤↔ t➼❝❤ tr➯♥ [a, b]✳ ỵ f tr [a, b] ✈➔ ❝â ❝❤➾ ♠ët sè ❤ú✉ ❤↕♥ ✤✐➸♠ ❣✐→♥ ✤♦↕♥ tr➯♥ ♦ t❤➻ f ❦❤↔ t➼❝❤ tr➯♥ [a, b] ỵ f ỡ ✈➔ ❜à ❝❤➦♥ tr➯♥ [a, b] t❤➻ f ❦❤↔ t➼❝❤ tr➯♥ [a, b]✳ [a, b] ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ tỉ✐ s➩ t ự ởt ỵ t t ữ ợ ởt tờ ữ trữợ t ú tổ ữ r ✤à♥❤ ♥❣❤➽❛ s❛✉✳ ❆✳✸ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ✣à♥❤ ♥❣❤➽❛ ❆✳✶✽✳ ✶✺✶ ✣÷í♥❣ ❦➼♥❤ ❝õ❛ ♣❤➙♥ ✤✐➸♠ P ❧➔ λ(P) = max{∆xi |i = 1, n} ✣à♥❤ ♥❣❤➽❛ ❆✳✶✾✳ ❝❤♦ ●✐↔ sû xi−1 ≤ ti ≤ xi ✱ i = 1, n✱ P ❧➔ ♠ët ♣❤➙♥ ✤✐➸♠ ♥➔♦ ✤â ❝õ❛ ✤♦↕♥ [a, b]✳ ❈❤å♥ ❝→❝ ✤✐➸♠ ti s❛♦ ✈➔ ①➨t tê♥❣ n S(P) = f (ti )∆xi ✭❆✳✸✼✮ i=1 ❚❛ ✤à♥❤ ♥❣❤➽❛ lim S(P ) = I, (P )0 ợ >0 tỗ t δ>0 λ(P) < δ s❛♦ ❝❤♦ ✈ỵ✐ t❛ ❝â |S(P) − I| < ε ✭❆✳✸✾✮ ❈➛♥ ♥â✐ rã t❤➯♠ r➡♥❣ ✈➳ ♣❤↔✐ ❝õ❛ ✭❆✳✸✼✮ ♥❣♦➔✐ ♣❤ö t❤✉ë❝ ✈➔♦ ti ✿ xi−1 ≤ ti ≤ xi ✱ i = 1, n sè ❝á♥ ♣❤ư t❤✉ë❝ ✈➔♦ ❜ë ❝→❝ ♥ú❛✱ ♥❤÷♥❣ ✤➸ ổ tự ù ỗ ú tổ ổ t S(P) sỹ tở tr ỵ P ỵ tỗ t lim S(P) ❦❤✐ λ(P) → t❤➻ ❤➔♠ f ❦❤↔ t➼❝❤ tr➯♥ [a, b] ✈➔ b lim S(P) = f (x)dx ✭❆✳✹✵✮ λ(P)→0 a ✭❜✮ ◆➳✉ ❤➔♠ t↕✐ f ❦❤↔ t➼❝❤ tr➯♥ [a, b] .rữợ t sỷ r ợ > s❛♦ ❝❤♦ ✈ỵ✐ λ(P) < δ t❛ ❝â t tọ tr tỗ t↕✐ ✈➔ ❜➡♥❣ I− ❈❤å♥ ♠ët ♣❤➙♥ ✤✐➸♠ P s❛♦ tr ú ữợ ú I ỵ f < S(P) < I + I✳ ●✐↔ sû ε > 0✳ ỗ (P) < ti ❝❤↕② tr➯♥ [xi−1 , xi ]✱ i = 1, n✱ sè S(P) ♥❤➟♥ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ➜② t❛ ✤✐ ✤➳♥ ✈➔ ❧➜② ❝➟♥ ≤ L(P) ≤ U (P) ≤ I + 2ε ❦❤↔ t➼❝❤ tr➯♥ [a, b] ✈➔ ✤✐➲✉ ✤â ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ ❦❤➥♥❣ ✤à♥❤ ✭❛✮ ✈➻ b L(P) ≤ f (x)dx ≤ U (P) a ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦❤➥♥❣ ✤à♥❤ ✭❜✮ ❣✐↔ sû r➡♥❣ f ❦❤↔ t➼❝❤ tr➯♥ b U (Q) < f (x)dx + a ✣➦t M = sup {f (x)|x ∈ [a, b]} ε [a, b] ỗ t Q s PP ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ❙➮ ✶✺✷ δ1 = 8Mε n ✱ tr♦♥❣ ✤â ♥ ❧➔ sè ✤♦↕♥ ♥❤ä ❝õ❛ ♣❤➙♥ ✤✐➸♠ Q✳ ●✐↔ sû P ❧➔ ♣❤➙♥ ✤✐➸♠ ❜➜t ❦ý✱ s❛♦ ❝❤♦ λ(P) < δ1 ✳ ❳➨t ❝→❝ tê♥❣ U (P), U (P ∪ Q)✳ ❍❛✐ tê♥❣ ➜② trị♥❣ ♥❤❛✉ tr➯♥ ❝→❝ ✤♦↕♥ ❝♦♥ ❝õ❛ P ❦❤ỉ♥❣ ❝❤ù❛ ❝→❝ ✤✐➸♠ ❝õ❛ Q ð ♣❤➼❛ tr♦♥❣✳ ❚r➯♥ ❝→❝ ✤♦↕♥ ❝♦♥ ❝õ❛ P ❝â ❝❤ù❛ ❝→❝ ✤✐➸♠ ❝õ❛ Q ð ♣❤➼❛ tr♦♥❣ ✤ë ❧➺❝❤ ❝õ❛ ❤❛✐ tê♥❣ ➜② ❝â ❣✐→ trà t✉②➺t ✤è✐ ❦❤ỉ♥❣ ✈÷đt q✉→ ❈❤å♥ (n−1)ε2M 8M n (n − 1) max ∆xi 2M < i=1,n < 4ε ❑➳t ❤đ♣ ✈ỵ✐ ✭❆✳✹✷✮ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ị♥❣ ❝❤♦ b U (P) < a ✈ỵ✐ ♠å✐ P t❤ä❛ ♠➣♥ ε f (x)dx + , λ(P) < δ1 ✳ ❈ô♥❣ ❜➡♥❣ ❝→❝❤ ➜② t❛ ❝â t❤➸ ❝❤ù♥❣ tỗ t b L(P) > a ợ t ✭❆✳✹✸✮ P t❤ä❛ ♠➣♥ λ(P) < δ2 ✳ δ = min(δ1 , δ2 ) t❛ t❤➜② ✭❆✳✹✸✮ δ2 > s❛♦ ❝❤♦ ε f (x)dx − , ✭❆✳✹✹✮ ✈➔ ✭❆✳✹✹✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ P s❛♦ ❝❤♦ λ(P) < δ ✳ ❱➻ rã r➔♥❣ ❧➔ L(P) ≤ S(P) ≤ U (P) ♥➯♥ tø ✭❆✳✹✸✮ ✈➔ ✭❆✳✹✹✮ s✉② r❛ b ε f (x)dx + , b ε f (x)dx − , S(P) < a S(P) > a ❝á♥ tø ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ ❞➵ t❤➜② b |S(P) − f (x)dx| < ε a ✈ỵ✐ ♠å✐ P s❛♦ ❝❤♦ λ(P) < δ ✣➸ ❦➳t t❤ó❝ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ ♥➯✉ ♠ët ✈➼ ❞ư ✤ì♥ ❣✐↔♥ →♣ ỵ • f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ➜②✳ ❉♦ ✤â ❤➔♠ f t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝ ✭❆✳✹✵✮✳ ❈❤å♥ ♣❤➙♥ ✤✐➸♠ ❝õ❛ ✤♦↕♥ [0, 1] ❧➔ P = { ni , n ∈ N∗ , i = 0, n} t❤ä❛ ♠➣♥ λ(P) = n1 → ❦❤✐ n → ∞✳ ▲➜② ti = ni ∈ [ i−1 , i ]✱ i = 1, n n n ỵ t t❤➜② ❤➔♠ ❚❛ ❝â S(P) = n i i=1 n n = f (x) = x n(n+1) 2n2 ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ = 12 (1 + n1 ), lim S(P) = lim [ 12 (1 + n1 )] = 12 λ(P)→0 ❱➟② xdx = 12 n→∞ [0, 1] ✈➻ ❆✳✸ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ✶✺✸ ❆✳✸✳✷✳✸✳ ❚➼♥❤ ❝❤➜t ❝õ❛ t➼❝❤ ♣❤➙♥ ①→❝ r t t ữợ t õ t❤✐➳t ❚➼♥❤ ❝❤➜t ✶ ✳ f ❦❤↔ t➼❝❤ tr➯♥ Cf (x)dx = C ✭❛✮ ◆➳✉ ❤➔♠ [a, b]✳ b a ❍ì♥ ♥ú❛ a < b✳ ✤♦↕♥ [a, b] t❤➻ ✈ỵ✐ ♠å✐ ❤➡♥❣ sè b f (x)dx a ✭❜✮ ◆➳✉ ❝→❝ ❤➔♠ f, g ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b] t❤➻ ❤➔♠ b b b ♥ú❛ [f (x) + g(x)]dx = a f (x)dx + a g(x)dx a ❚➼♥❤ ❝❤➜t ✷ c ∈ (a, b)✳ ✳ ◆➳✉ ❤➔♠ f [a, b] ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ C f +g ❤➔♠ Cf ❝ô♥❣ ❦❤↔ t➼❝❤ tr➯♥ ❝ô♥❣ ❦❤↔ t➼❝❤ tr➯♥ t❤➻ ❦❤↔ t➼❝❤ tr➯♥ [a, c] ✈➔ [a, b]✳ [c, b] ❍ì♥ ✈ỵ✐ ♠å✐ ❍ì♥ ♥ú❛ ✳ ✭❛✮ ◆➳✉ ❤➔♠ f ❦❤↔ t➼❝❤ tr➯♥ ✭❜✮ ◆➳✉ ❝→❝ ❤➔♠ f, g c a [a, b]✱ f (x) ≥ 0, ∀x ∈ [a, b]✱ a ❦❤↔ t➼❝❤ tr➯♥ g(x)dx [a, b] t❤➻ |f | ❍ì♥ ♥ú❛ b a ❦❤↔ t➼❝❤ tr➯♥ [a, b]✳ ❦❤↔ t➼❝❤ tr➯♥ |f (x)| dx f (x)dx ≤ f t❤➻ a b ✭❞✮ ◆➳✉ ❤➔♠ f (x)dx ≥ b f (x)dx ≥ f b a t❤➻ [a, b]✱ f (x) ≥ g(x), ∀x ∈ [a, b]✱ ❦❤↔ t➼❝❤ tr➯♥ b ✭❝✮ ◆➳✉ ❤➔♠ f (x)dx f (x)dx + f (x)dx = a ❚➼♥❤ ❝❤➜t ✸ b c b a [a, b]✱ m ≤ f (x) ≤ M, ∀x ∈ [a, b]✱ t❤➻ b m(b − a) ≤ f (x)dx ≤ M (b − a) a ❚➼♥❤ ❝❤➜t ✹ ✳ ✭❛✮ ỵ tr tự t t t tỗ t [m, M ] f t tr➯♥ [a, b]✱ m ≤ f (x) ≤ M, ∀x ∈ [a, b]✱ s❛♦ ❝❤♦ b f (x)dx = µ(b − a) a ✣➦❝ ❜✐➺t ♥➳✉ ❤➔♠ f ❧✐➯♥ tö❝ tr [a, b] t tỗ t c [a, b] s❛♦ ❝❤♦ b f (x)dx = f (c)(b − a) a ỵ tr tự ❤➔♠ ❦❤ỉ♥❣ ❞÷ì♥❣ tr➯♥ [a, b]✱ m ≤ f (x) ≤ M, ∀x ∈ [a, b]✱ f, g b a t [a, b] t tỗ t g(x)dx a f ❧✐➯♥ tö❝ tr➯♥ [a, b]✱ g c ∈ [a, b] s❛♦ ❝❤♦ ❦❤↔ t➼❝❤ tr➯♥ b [a, b]✱ g b f (x)dx = f (c) a ❤♦➦❝ b f (x)g(x)dx = µ tr➯♥ [a, b]✱ g ❦❤ỉ♥❣ ➙♠ µ ∈ [m, M ] s❛♦ ❝❤♦ ❦❤↔ t➼❝❤ tr➯♥ t t tỗ t g(x)dx a ổ ổ ❞÷ì♥❣ P❍➆P ❚➑◆❍ ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ỵ t tr ❈æ♥❣ t❤ù❝ ◆❡✇t♦♥✲▲❡✐❜♥✐③ ●✐↔ sû f ✶✳✸✳✷✳✸✮ a 0, x0 + h ≤ b t❤❡♦ ❤➔♠ ●✐↔ sû ✭❆✳✹✻✮ s❛♦ ❝❤♦ m ≤ f (x) ≤ M, ∀x ∈ [a, b]✳ t➼♥❤ ❝❤➜t ✷ ❝õ❛ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ✭♠ö❝ ✶✳✸✳✷✳✸✮ x0 +h Φ(x0 + h) − Φ(x0 ) = x0 +h x0 f (t)dt − f (t)dt = f (t)dt x0 a a ❞♦ ✤â t❤❡♦ t➼♥❤ ❝❤➜t ✹ ♣❤➛♥ ✭❛✮ ❝õ❛ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ✭♠ö❝ tỗ t s (x0 + h) Φ(x0 ) = µh ➊ ➍ µ∈ inf f (t), t∈[x0 ,x0 +h] ❈❤♦ ❤➔♠ Φ h → 0+ ✳ ❱➻ µ ❇➙② ❣✐í ❣✐↔ sû f µh → 0✱ ✭❆✳✹✽✮ h → 0+ ✳ µ ❱➻ ❧✐➯♥ tư❝ t↕✐ f xo ✳ ❧✐➯♥ tư❝ t↕✐ ❝ơ♥❣ ❞➛♥ ✤➳♥ ✭❆✳✹✽✮ s✉② r❛ Φ(x0 + h) − Φ(x0 ) → 0✱ tù❝ ❧➔ xo ✳ ❚ø ✭❆✳✹✼✮ s✉② r❛ Φ(x0 +h)−Φ(x0 ) h ❈❤♦ f (t) ⊂ [m, M ] sup t∈[x0 ,x0 +h] ❜à ❝❤➦♥ ❞♦ ✭❆✳✹✽✮ ♥➯♥ ❧✐➯♥ tö❝ ♣❤↔✐ t↕✐ ✭❆✳✹✼✮ f (x0 )✱ xo ♥➯♥ inf = µ f (t) t∈[x0 ,x0 +h] sup ✈➔ f (t) ✤➲✉ ❞➛♥ ✤➳♥ f (x0 )✳ ❉♦ t∈[x0 ,x0 +h] tù❝ ❧➔ lim h→0+ Φ(x0 +h)−Φ(x0 ) h = f (x0 ) Φ+ (x0 ) = f (x0 )✳ ●✐↔ sû x0 ∈ (a, b]✳ ▼ët ❝→❝❤ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü ❝â t❤➸ ❝❤ù♥❣ tä ❤➔♠ Φ ❧✐➯♥ tư❝ tr→✐ t↕✐ xo ✈➔ ♥➳✉ f ❧✐➯♥ tö❝ t↕✐ xo t❤➻ Φ− (x0 ) = f (x0 )✳ ✣✐➲✉ ✤â ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ ❝↔ ❤❛✐ ♣❤➛♥ ✭❛✮ ✈➔ ✭❜✮ ỵ t f (x), x [a, b] ứ ỵ s r ♥➳✉ ❤➔♠ ❈â ♥❣❤➽❛ ❧➔ Φ f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ f tr➯♥ [a, b] t (x) = [a, b] ỵ ●✐↔ sû a < b✳ ◆➳✉ ❤➔♠ f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b] ✈➔ F ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ f tr➯♥ [a, b] t❤➻ ♦ b f (x)dx = F (b) − F (a) a ❆✳✸ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ❍✐➺✉ sè F (b) − F (a) tữớ ữủ ỵ F (x)|ba ✱ ❞♦ ✤â ❝æ♥❣ t❤ù❝ tr➯♥ ❝á♥ ❝â ❞↕♥❣ b f (x)dx = F (x)|ba ✭❆✳✹✾✮ a ❈æ♥❣ t❤ù❝ ✭❆✳✹✾✮ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ◆❡✇t♦♥✲▲❡✐❜♥✐③✳ ∆.❱➻ F f ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ ❝õ❛ ❤➔♠ f tr➯♥ [a, b] tr➯♥ [a, b] ✈➔ t❤❡♦ ♥❤➟♥ ①➨t ❆✳✼ t❤➻ Φ ❝ô♥❣ ❧➔ ♥❣✉②➯♥ ♥➯♥ x f (t)dt = F (x) + C, ∀x ∈ [a, b] Φ(x) = a ❈❤♦ x=a t❛ ✤÷đ❝ = Φ(a) = F (a) + C ⇒ C = −F (a)✳ ❉♦ ✤â x f (t)dt = F (x) − F (a), ∀x ∈ [a, b] a ❈✉è✐ ❝ị♥❣ ❝❤♦ • ❱➼ ❞ư ❆✳✶✹✳ ❱➻ x=b x2 t❛ ✤÷đ❝ ❝ỉ♥❣ t❤ù❝ ✭❆✳✹✾✮ ❧➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ xdx = x2 x = tr➯♥ ✤♦↕♥ [0, 1] ♥➯♥ t❤❡♦ ❝æ♥❣ t❤ù❝ ✭❆✳✹✾✮ 12 02 − = 2 ❚❛ t❤➜② ❧↕✐ ❦➳t q✉↔ ❝õ❛ ✈➼ ❞ư ❆✳✶✸✳ ❆✳✸✳✷✳✺✳ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ❆✳✸✳✷✳✺✳✶✳ P❤÷ì♥❣ ♣❤→♣ ✤ê✐ ❜✐➳♥ a ❚r♦♥❣ ❝→❝ ✤à♥❤ ỵ b số tỹ tọ a < b ỵ ❜✐➳♥ ❞↕♥❣ ✶✮✳ ●✐↔ sû ✭✐✮ ❍➔♠ f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b]❀ ✭✐✐✮ ❍➔♠ u(x) ❦❤↔ ✈✐ tr➯♥ [a, b] u (x) ổ ỗ t tr [a, b]✱ ✈➔ f (x)dx g(u(x))u (x)dx, ∀x ∈ [a, b]❀ ✭✐✐✐✮ g(t) ❧✐➯♥ tö❝ tr➯♥ t➟♣ I = {u(x)|x ∈ [a, b]}✳ ❑❤✐ ✤â b u(b) f (x)dx = a ∆.❱➻ g(t)dt u(a) I ♥➯♥ ❝â ♥❣✉②➯♥ ❤➔♠ tr➯♥ t➟♣ ➜②✳ ●✐↔ sû G ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ❝õ❛ g tr➯♥ I ✳ ✣➦t F (x) = G(u(x))✳ ❈❤ù♥❣ tữỡ tỹ ữ ỵ õ t t F ❧➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ f tr➯♥ [a, b]✳ ✣✐➸♠ ❦❤→❝ ð ✤➙② ❧➔ ❦❤➥♥❣ ✤à♥❤ F (x) = f (x) t↕✐ x = a ✈➔ x = b✳ ❚❛ tứ ỵ g = ❧✐➯♥ tö❝ tr➯♥ t➟♣ b f (x)dx = F (b) − F (a) = G(u(b)) − G(u(a)), a u(b) g(t)dt = G(u(b)) − G(u(a)) u(a) ❚ø ❝→❝ ✤➥♥❣ tự tr t ữủ ỵ P❍➆P ❚➑◆❍ ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ỵ sû ✭✐✮ ❍➔♠ f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b]❀ ✭✐✐✮ ❍➔♠ ϕ(t) ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [α, β]❀ ✭✐✐✐✮ ϕ(α) = a, ϕ(β) = b❀ ✭✐✈✮ ϕ(t) ∈ [a, b], ∀t ∈ [α, β]✳ ❑❤✐ ✤â b β ✶✺✻ f (x)dx = f (ϕ(t))ϕ (t)dt a α ∆.●✐↔ sû F (x) ❧➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ f (x) f ((t)) (t) tr [, ] ỵ tr➯♥ [a, b]✳ ❑❤✐ ✤â F (ϕ(t)) ❧➔ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ b f (x)dx = F (b) − F (a), a β f (ϕ(t))ϕ (t)dt = F (ϕ(β)) − F (ϕ(α)) = F (b) − F (a) α ❚ø ❝→❝ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ ❦❤➥♥❣ ✤à♥❤ ❝õ❛ ỵ ã t dt ✳ ❚❛ + t2 π ❧✐➯♥ tö❝ tr➯♥ t➟♣ [0, 1]✱ ❧➔ ↔♥❤ ❝õ❛ ✤♦↕♥ [0, ] q✉❛ ❤➔♠ sinx✳ ❱ỵ✐ x = t❤➻ t = 0✱ t = sin x ⇒ dt = cos xdx✳ 1+t2 t❤➜② ❤➔♠ x= cos xdx + sin2 x π/2 ✭❛✮ ❚➼♥❤ t❤➻ t = 1✳ π/2 õ tự ữợ t tr t ỵ cos xdx = + sin2 x √ = arctan t|10 = arctan − arctan = dt 1+t2 = dx (1+x2 )3 = π/4 dt cos2 t.1/cos3 t π dx x = tan t, ≤ t ≤ π4 , ⇒ dx = cosdt2 t ✳ ❍➔♠ cos12 t ❧✐➯♥ tö❝ x = ⇒ t = π4 ✱ tan t ∈ [0, 1], ∀t [0, ] ỵ 0= (1+x2 )3 ✣➦t π π/4 √ dt cos2 t (1+tan2 t)3 = π/4 cos tdt = = π/4 sin t|π/4 dt cos2 t (1/cos2 t) = tr➯♥ [0, π4 ]✳ ❚❛ ❝â x = ⇒ t = 0✱ π/4 dt cos2 t|1/cos3 t| √ √ 2 = sin(π/4) − sin = −0= 2 ❆✳✸✳✷✳✺✳✷✳ P❤÷ì♥❣ ♣❤→♣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ◆➳✉ ❝→❝ ❤➔♠ u, v ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [a, b] t❤➻ t❛ ❝â ❝æ♥❣ t❤ù❝ s❛✉ ❣å✐ ❧➔ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ b b udv = (uv)|ba − a ❱➼ ❞ö ❆✳✶✻✳ • vdu ✭❆✳✺✵✮ a ❚➼♥❤ π/2 sinn xdx, In = ✭❆✳✺✶✮ ❆✳✸ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❤➔♠ ♠ët ❜✐➳♥ ✶✺✼ π/2 cosn xdx, Jn = ✭❆✳✺✷✮ ✈ỵ✐ n ∈ N✳ ❚❛ ❝â π/2 dx = x|π/2 = π2 , I0 = π/2 sin xdx = − cos x|π/2 = I1 = ❱ỵ✐ n≥2 t❛ ❝â π/2 π/2 π/2 In = sinn−1 xd(− cos x) = − sinn−1 x cos x + cos xd(sinn−1 x) = π/2 π/2 = (n − 1) sinn−2 xcos2 xdx = (n − 1) sinn−2 x(1 − sin2 x)dx = π/2 = (n − 1) (sinn−2 x − sinn x)dx = (n − 1)In−2 − (n − 1)In ❙✉② r❛ In = n−1 In−2 n 2m − 2m − I0 , 2m 2m − 2 ợ m N ỵ t➼❝❤ ❝→❝ sè ❧➫ ❧✐➯♥ t✐➳♣ tø ✶ ✤➳♥ 2m − ❧➔ (2m − 1)!! ✭✤å❝ ❧➔ 2m − ❣✐❛✐ t❤ø❛ ❝→❝❤✮✱ t➼❝❤ ❝→❝ sè ❝❤➤♥ ❧✐➯♥ t✐➳♣ tø ✷ ✤➳♥ 2m ❧➔ (2m)!! ✭✤å❝ ❧➔ 2m ❣✐❛✐ t❤ø❛ ❝→❝❤✮✱ ✈➔ π (2m − 1)!! π t❤❛② I0 = t❛ ✤÷đ❝ I2m = (2m)!! (2m)!! ữỡ tỹ I2m+1 = ợ m N (2m + 1)!! t tữỡ tỹ ợ Jn t❛ ✤✐ ✤➳♥ ❚ø ✤â ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t❛ ♥❤➟♥ ✤÷đ❝ I2m = J2m = (2m−1)!! π , I2m+1 (2m)!! I2m = (2m)!! ✈ỵ✐ ♠å✐ (2m+1)!! = J2m+1 = m ∈ N∗ ❆✳✸✳✸✳ ❚➼❝❤ ♣❤➙♥ s✉② rë♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝➟♥ ❧➜② t➼❝❤ ♣❤➙♥ ❧➔ ✈ỉ ❤↕♥ ●✐↔ sû ❤➔♠ f ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b] ✈ỵ✐ ♠å✐ b t❤ä❛ ♠➣♥ b > a✳ ◆➳✉ tỗ t ợ b lim f (x)dx b+ a f t ợ õ ữủ t s✉② rë♥❣ ❝õ❛ ❤➔♠ ❧➔ +∞ f (x)dx f (x)dx = lim f [a, +) ữủ ỵ b b→+∞ a ❚÷ì♥❣ tü✱ ♥➳✉ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ ✭❆✳✺✸✮ a [a, b] ✈ỵ✐ ♠å✐ ❛ t❤ä❛ a < b tỗ t ợ b lim a f (x)dx a t ợ õ ữủ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❝õ❛ ❤➔♠ ❧➔ b tr (, b] ữủ ỵ b f (x)dx = lim −∞ f a→−∞ f (x)dx a ✭❆✳✺✹✮ P❍➆P ❚➑◆❍ ❱■✱ ❚➑❈❍ P❍❹◆ ❍⑨▼ ❙➮ ▼❐❚ ❇■➌◆ ❙➮ ✶✺✽ ◆➳✉ ✈ỵ✐ ♠å✐ ❛ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❝õ❛ ❤➔♠ t❤➻ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❝õ❛ ❤➔♠ f tr➯♥ ❦❤♦➔♥❣ +∞ f tr➯♥ ❝→❝ ❦❤♦↔♥❣ (−∞, a] (−∞, +∞) a [a, +) tỗ t + f (x)dx = f (x)dx + −∞ ✈➔ −∞ f (x)dx ✭❆✳✺✺✮ a ❑❤✐ ✈➳ tr→✐ ❝õ❛ ❝→❝ ❝æ♥❣ t❤ù❝ ✭❆✳✺✸✮ ✲ ✭❆✳✺✺✮ tỗ t t t õ t s rở ➜② ❤ë✐ tư✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥❣÷đ❝ ❧↕✐ t❛ ♥â✐ ❝❤ó♥❣ ♣❤➙♥ ❦ý✳ ❱➼ ❞ư ❆✳✶✼✳ • ✭❛✮ ❚❛ ❝â +∞ dx 1+x2 b dx b→+∞ 1+x dx −∞ 1+x2 ✭❜✮ ❚✐➳♣ t❤❡♦ = lim [arctan x|b0 ] = lim arctan b = = lim = b→+∞ dx lim a 1+x a→−∞ b→+∞ π = lim [arctan x|0a ] = lim [− arctan a] = a→−∞ a→−∞ π ✭❝✮ ❚ø ❦➳t q✉↔ ❝õ❛ ❝→❝ ♣❤➛♥ ✭❛✮ ✈➔ ✭❜✮✱ ✈➔ ❝æ♥❣ t❤ù❝ ✭❆✳✺✺✮ t❛ ❝â +∞ −∞ −∞ ✭❞✮ ❳➨t xex dx✳ dx 1+x2 = −∞ a dx 1+x2 = π π + = π 2 x xe dx = + ❚❛ ❝â x +∞ dx 1+x2 xde = xex |0a a ex dx = −aea − ex |0a = −aea − + ea = ea (1 − a) − − a ⑩♣ ❞ö♥❣ q✉② t➢❝ ▲✬❍♦s♣✐t❛❧ t❛ ❝â lim [ea (1 − a) − 1] = lim 1−a −a a→−∞ e a→−∞ ❱➟② −∞ a→−∞ a xex dx = lim xex dx = −1 −1 −a a→−∞ −e − 1= lim − 1= lim ea − = −1 a→−∞ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ P❤↕♠ ❱➠♥ ▼✐♥❤✱ ◆❣✉②➵♥ ❚❤à ❍➡♥❣✱ P❤↕♠ ❚❤✉ ❍♦➔✐✱ ❇➔✐ ❣✐↔♥❣ ❣✐↔✐ t➼❝❤ ✳ ❚➔✐ ❧✐➺✉ ❧÷✉ ❤➔♥❤ ♥ë✐ ❜ë✱ ✣↕✐ ❤å❝ ❍➔♥❣ ❍↔✐ ❱✐➺t ◆❛♠✱ ✷✵✶✷✳ ❬✷❪ ❯✳ ❘✉❞✐♥✱ ♦s♥♦✈② ♠❛t❡♠❛t✐q❡s❦♦❣♦ ❛♥❛❧✐③❛✱ ✐③❞❛t❡❧⑦st✈♦ ✧♠✐r✧✱ ▼♦s❦✈❛ ✶✾✼✻✳