❈❤➢➡♥❣ ✷ P❤Ð♣ tÝ♥❤ ✈✐ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ✷✳✶ ✷✳✶✳✶ ➜➵♦ ❤➭♠ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✶✳ ❈❤♦ ❤➭♠ sè r➺♥❣ ❤➭♠ sè f (x) f (x) ①➳❝ ➤Þ♥❤ tr♦♥❣ ❦❤♦➯♥❣ ❝ã ➤➵♦ ❤➭♠ t➵✐ ➤✐Ó♠ c ∈ (a, b) (a, b)✳ ❚❛ ♥ã✐ ♥Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ f (x) − f (c) x→c x−c A = lim ❑❤✐ ➤ã sè A ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➵♦ ❤➭♠ ❝đ❛ ❤➭♠ sè f (x) t➵✐ x = c ✈➭ ❦ý ❤✐Ö✉ A = f (c)✳ ◆Õ✉ ❤➭♠ sè f (x) ❝ã ➤➵♦ ❤➭♠ t➵✐ ♠ä✐ ➤✐Ĩ♠ x ∈ (a, b) t❤× ❤➭♠ f ❝ã ➤➵♦ ❤➭♠ tr➟♥ ❦❤♦➯♥❣ (a, b)✳ ◆❤❐♥ ①Ðt✳ ✭✶✮ ◆Õ✉ ➤➷t x = c + x t❤× f (c) = lim x→0 f (c + x) − f (c) x ✺✽ P❤Ð♣ tÝ♥❤ ✈✐ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ❚r♦♥❣ ➤ã✱ x ❣ä✐ ❧➭ sè ❣✐❛ ❝ñ❛ ➤è✐ sè✱ f (c + x) − f (c) ❧➭ sè ❣✐❛ ❝ñ❛ ❤➭♠ sè t➵✐ x = c✳ ✭✷✮ ❚❛ ❝ã t❤Ó ✈✐Õt ❝➠♥❣ t❤ø❝ ➤➵♦ ❤➭♠ ❝đ❛ ❤➭♠ sè ❞➢í✐ ❞➵♥❣ s❛✉ f (c + x) − f (c) = f (c) x + o( x) ❱Ý ❞ô✳ ✭✶✮ f (x) = c, x ∈ (a, b) ⇒ f (x) = ✈× f (x + x) − f (x) = c − c = 0✳ ✭✷✮ f (x) = xn , x ∈ (a, b)( ✈í✐ n ∈ N \ {0}) ⇒ f (x) = nxn−1 ✈× x)n − xn x→0 x n−1 + Cn2 xn−2 ( x) + · · · + Cnn ( x)n−2 ) lim (Cn x f (x) = lim = (x + x→0 Cn1 xn−1 = = nxn−1 ✭✸✮ f (x) = sin x, x ∈ (a, b) ⇒ f (x) = cos x✳ ❚❤❐t ✈❐②✱ t❛ ❝ã f (x + x) − f (x) = sin(x + x) − sin x = sin x x cos(x + ) 2 x cos(x + x ) = cos x ⇒ f (x) = lim x x→0 2 ✭✹✮ f (x) = ln x, x ∈ (a, b) ⊂ R+ ⇒ f (x) = ✈× x sin f (x) = lim x→0 ý ln(x + x) − ln x = lim x x x→0 x ) x = x x x ln(1 + ♥❣❤Ü❛ ❤×♥❤ ❤ä❝ ❝đ❛ ➤➵♦ ❤➭♠ ➜➵♦ ❤➭♠ ❝đ❛ ❤➭♠ sè f (x) t➵✐ ➤✐Ĩ♠ x0 ❧➭ ❤Ư sè ❣ã❝ ❝đ❛ t✐Õ♣ t✉②Õ♥ ✈í✐ ➤➢ê♥❣ ❝♦♥❣ y = f (x) t➵✐ ➤✐Ó♠ M (x0 , f (x0 )✳ ✺✾ ✷✳✶ ➜➵♦ ❤➭♠ ◆❤❐♥ ①Ðt✳ ✭✶✮ ◆Õ✉ ❤➭♠ sè f (x) ❝ã ➤➵♦ ❤➭♠ t➵✐ ➤✐Ĩ♠ x = c t❤× ➤å t❤Þ ❝đ❛ ❤➭♠ sè f (x) ❝ã ♠ét t✐Õ♣ t✉②Õ♥ ❞✉② ♥❤✃t t➵✐ (c, f (c) ❦❤➠♥❣ ✈✉➠♥❣ ❣ã❝ ✈í✐ trơ❝ Ox✳ ✭✷✮ ◆Õ✉ ❤➭♠ sè f (x) ❝ã ➤➵♦ ❤➭♠ t➵✐ ➤✐Ĩ♠ x = c t❤× ♥ã ❧✐➟♥ tơ❝ t➵✐ x = c ✭➤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ➤ó♥❣✮✳ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✷✳ ●✐➯ sư ❤➭♠ sè f (x) ①➳❝ ➤Þ♥❤ tr➟♥ ❦❤♦➯♥❣ (a, b), x0 ∈ (a, b)✳ ◆Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ lim + f (x0 + x→0 t❤× f ❝ã ➤➵♦ ❤➭♠ ♣❤➯✐ t➵✐ x0 x) − f (x0 ) x ✈➭ ❦ý ❤✐Ö✉ f+ (x0 ) = f (x0 + 0) = lim + f (x0 + x→0 x) − f (x0 ) x ❚➢➡♥❣ tù✱ ♥Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ ữ lim f (x0 + x0 tì f ❝ã ➤➵♦ ❤➭♠ tr➳✐ t➵✐ x0 x) − f (x0 ) x ✈➭ ❦ý ❤✐Ö✉ f− (x0 ) = f (x0 − 0) = lim − f (x0 + x→0 ❱Ý ❞ô✳ x) − f (x0 ) x ❳Ðt ❤➭♠ f (x) = |x| ❚❛ ❝ã f+ (0) = ✈➭ f− (0) = −1✳ ◆❤❐♥ ①Ðt✳ ✭✶✮ ➜➵♦ ❤➭♠ f (x0 ) tå♥ t➵✐ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ f+ (x0 ), f− (x0 ) ❝ï♥❣ tå♥ t➵✐ ✈➭ ❜➺♥❣ ♥❤❛✉✳ ❑❤✐ ➤ã t❛ ❝ã f (x0 ) = f+ (x0 ) = f− (x0 ) x) − f (x0 ) = ±∞ t❤× t❛ ♥ã✐ t➵✐ ➤✐Ĩ♠ x = x0 x→0 x ❤➭♠ sè f (x) ❝ã ➤➵♦ ❤➭♠ ✈➠ ❝ï♥❣ ✭❦❤✐ ➤ã t✐Õ♣ t✉②Õ♥ t➵✐ x0 ✈✉➠♥❣ ❣ã❝ ✈í✐ trơ❝ ❤♦➭♥❤✮✳ ✭✷✮ ◆Õ✉ lim f (x0 + ✻✵ P❤Ð♣ tÝ♥❤ ✈✐ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ✷✳✶✳✷ ❈➳❝ q✉② t➽❝ tÝ♥❤ ➤➵♦ ❤➭♠✳ ➜➵♦ ❤➭♠ ❝đ❛ ❤➭♠ sè ❤ỵ♣✱ ➤➵♦ ❤➭♠ ❝đ❛ ❤➭♠ sè ♥❣➢ỵ❝ f (x) ✈➭ g(x) ❧➭ ❤❛✐ ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr♦♥❣ ❦❤♦➯♥❣ (a, b) ✈➭ ❝ã ➤➵♦ ❤➭♠ t➵✐ x ∈ (a, b)✳ ❑❤✐ ➤ã ✭✶✮ (f (x) ± g(x)) = f (x) ± g (x) ✭✷✮ (Cf (x)) = Cf (x) ✭✸✮ (f (x)g(x)) = f (x)g(x) + f (x).g (x) f (x) f (x)g(x) − g (x)f (x) ✈í✐ g(x) = ✭✹✮ = g(x) g (x) ➜Þ♥❤ ❧Ý ✷✳✶✳✸✳ ❈❤♦ ❈❤ø♥❣ ♠✐♥❤✳ ❈➳❝ ❝➠♥❣ t❤ø❝ ✭✶✮✱ ✭✷✮ s✉② trù❝ t✐Õ♣ tõ tÝ♥❤ ❝❤✃t ❝đ❛ ❣✐í✐ ❤➵♥ ✈➭ ➤Þ♥❤ ♥❣❤Ü❛ ➤➵♦ ❤➭♠✳ ✭✸✮ ❚❛ ❝ã (f (x)g(x)) = lim f (x + x→0 x) − f (x)g(x) x)g(x + x x)g(x + x) − f (x + x→0 x (f (x + x)g(x) − f (x)g(x)) + x = f (x)g (x) + f (x)g(x) = lim ✭✹✮ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ g(x) = lim x→0 (f (x + g(x) g(x + = x) x g(x)g(x + −g (x) = g (x) = lim x→0 −g (x) g (x) − x)g(x)) ❚❤❐t ✈❐②✱ g(x) x) −(g(x + x) − g(x)) x ❚õ ❦Õt q✉➯ tr➟♥ ❦Õt ❤ỵ♣ ✈í✐ ✭✸✮ t❛ ❝ã ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✻✶ ✷✳✶ ➜➵♦ ❤➭♠ ➜Þ♥❤ ❧Ý ✷✳✶✳✹✳ ●✐➯ sư u = g(x) ①➳❝ ➤Þ♥❤ tr♦♥❣ ❦❤♦➯♥❣ (a, b)✱ ❧✃② ❣✐➳ trÞ tr♦♥❣ (c, d) ✈➭ ❝ã ➤➵♦ ❤➭♠ t➵✐ e ∈ (a, b) ✭✷✮ ❍➭♠ y = f (x) ①➳❝ ➤Þ♥❤ tr♦♥❣ ❦❤♦➯♥❣ (c, d) ✈➭ ❝ã ➤➵♦ ❤➭♠ t➵✐ u = g(e)✳ ❑❤✐ ➤ã ❤➭♠ ❤ỵ♣ f◦ g ❝ã ➤➵♦ ❤➭♠ t➵✐ e ✈➭ ✭✶✮ ❍➭♠ sè (f◦ g(e)) = fu (g(e))g (e) ❈❤ø♥❣ ♠✐♥❤✳ ❉♦ f ❝ã ➤➵♦ ❤➭♠ t➵✐ f (u + ▼➷t ❦❤➳❝ g u ♥➟♥ u) − f (u) = f (u) u + o( u) ❝ã ➤➵♦ ❤➭♠ t➵✐ e ♥➟♥ u = g(e + x) − g(e) = g (e) x + o( x) u ✈➭♦ ❝➠♥❣ t❤ø❝ tr➟♥ t❛ ❝ã ❚❤❛② f (u + u) − f (u) = fu (g(e)).g (e) x + fu (u)o( x) + o( u) ❈❤✐❛ ❝➯ ❤❛✐ ✈Õ ❝❤♦ x → t❤× x ✈➭ ❝❤ó ý✱ o( x) →0 x ✈➭ g(x) ❧✐➟♥ tô❝ t➵✐ e u → 0✳ ❚❛ ❝ã (f◦ g(e)) = fu (g(e))g (e) ❱Ý ❞ô✳ ✭✶✮ f (x) = sin(ln x) ⇒ f (x) = cos(ln x) ✳ ❚❤❐t ✈❐② t❛ ❝ã x f (x) = sin u ✈í✐ u = ln x ❚❤❡♦ q✉② t➽❝ ❧✃② ➤➵♦ ❤➭♠ ❝đ❛ ❤➭♠ ❤ỵ♣✱ t❛ ❝ã f (x) = (sin u)u ux = cos u cos(ln x) = x x ✭✷✮ f (x) = loga (x2 + 1)(a > 0, a = 1) ⇒ f (x) = (x2 + 1) ln a ♥➟♥ ❦❤✐ ✻✷ P❤Ð♣ tÝ♥❤ ✈✐ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ❚❤❐t ✈❐②✱ t❛ ❝ã loga (x2 + 1) = loga e ln(x2 + 1) = ln u , ✈í✐ u = x2 + ln a ❉♦ ➤ã f (x) = (ln u)u ux 2x = 2x = ln a u ln a (x + 1) ln a ✭✸✮ ❚õ ➤➵♦ ❤➭♠ ❝ñ❛ ❤➭♠ sè y = sin x t❛ ❝ã (cos x) = − sin x, (tan x) = −1 , ( ❝♦t x) = cos x sin2 x ➜Þ♥❤ ❧Ý ✷✳✶✳✺✳ ✭➜➵♦ ❤➭♠ ❝đ❛ ❤➭♠ ♥❣➢ỵ❝✮✳ ●✐➯ sư f : [a, b] −→ [c, d] ❧➭ ♠ét g = f −1 : [c, d] −→ [a, b] ❧➭ ❤➭♠ ♥❣➢ỵ❝ ❝đ❛ ♥ã✳ ◆Õ✉ f ❝ã ➤➵♦ ❤➭♠ t➵✐ x0 ∈ [a, b] ✈➭ f (x0 ) = t❤× g ❝ã ➤➵♦ ❤➭♠ t➵✐ y0 = f (x0 ) ✈➭ gy (y0 ) = ✳ f (x0 ) s♦♥❣ ➳♥❤✱ ❧✐➟♥ tô❝ ✈➭ ❈❤ø♥❣ ♠✐♥❤✳ ◆Õ✉ y ∈ [c, d], y = y0 t❛ ❝ã x = g(y) = g(y0 ) = x0 ✱ ❤❛② x = x0 ✳ ❑❤✐ ➤ã g(y) − g(y0 ) x − x0 = = f (x) − f (x0 ) y − y0 f (x) − f (x0 ) x − x0 y → y0 t❤× g(y) → g(y0 ) ❞♦ g ❧✐➟♥ tô❝✳ ❉♦ ➤ã x → x0 t❛ ❝ã f (x) − f (x0 ) → f (x0 )✱ tõ ➤ã s✉② r❛ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ x − x0 π π ❱Ý ❞ô✳ ❳Ðt ❤❛✐ f (x) = sin x tr➟♥ [− , ] ✈➭ g(y) = arcsin y tr➟♥ [−1, 1]✳ 2 π π ❱í✐ x0 ❜✃t ❦ú t❤✉é❝ [− , ]✱ y0 = sin x0 ✈➭ f (x0 ) = cos x0 = 2 − y02 1 ❚❤❡♦ ➤Þ♥❤ ❧ý tr➟♥✱ gy (y0 ) = = f (x0 ) − y02 ❑❤✐ ❇➯♥❣ ➤➵♦ ❤➭♠ ❝ñ❛ ♠ét sè ❤➭♠ sè s➡ ❝✃♣ ✭✶✮ y = c ⇒ y = ✭✷✮ y = loga x ⇒ y = loga e ✳ ➜➷❝ ❜✐Ưt✱ ♥Õ✉ y = ln x t❤× y = x x ✻✸ ✷✳✷ ❱✐ ♣❤➞♥ ✭✸✮ y = ax ⇒ y = ax ln a ✭✹✮ y = xα (α ∈ R, x = 0) ⇒ y = αxα−1 ✭✺✮ y = sin x ⇒ y = cos x ✭✻✮ y = cos x ⇒ y = − sin x ✭✼✮ y = tan x ⇒ y = cos2 x ✭✽✮ y = ❝♦t x ⇒ y = − sin x ✭✾✮ y = arcsin x ⇒ y = √ − x2 ✭✶✵✮ y = arccos x ⇒ y = − √ − x2 ✭✶✶✮ y = arctan x ⇒ y = + x2 ✭✶✷✮ y = arccotx ⇒ y = − ✳ + x2 ✷✳✷ ❱✐ ♣❤➞♥ ✷✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ✈✐ ♣❤➞♥✱ ❤➭♠ sè ❦❤➯ ✈✐ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✶✳ ●✐➯ sö f f ❧➭ ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr♦♥❣ (a, b)✳ ❑❤✐ ➤ã ❤➭♠ sè x0 ∈ (a, b) ♥Õ✉ tå♥ t➵✐ sè A ∈ R x = x − x0 ✱ sè ❣✐❛ f (x0 ) = f (x) − f (x0 ) ❝ã t❤Ĩ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➯ ✈✐ ✭❝ã ✈✐ ♣❤➞♥✮ t➵✐ ➤✐Ó♠ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ sè ❣✐❛ ❜✐Ĩ✉ ❞✐Ơ♥ ➤➢ỵ❝ ❞➢í✐ ❞➵♥❣ f (x0 ) = A x + o( x) ❇✐Ó✉ t❤ø❝ A x ❣ä✐ ❧➭ ✈Þ ♣❤➞♥ ❝đ❛ ❤➭♠ sè f t➵✐ ➤✐Ĩ♠ x0 ✈➭ ❦ý ❤✐Ư✉ d(f (x0 )✳ ◆Õ✉ f ❦❤➯ ✈✐ t➵✐ ♠ä✐ ➤✐Ó♠ x ∈ (a, b) t❤× t❛ ♥ã✐ f ❦❤➯ ✈✐ tr♦♥❣ ❦❤♦➯♥❣ (a, b) ➜Þ♥❤ ❧Ý ✷✳✷✳✷✳ ❍➭♠ sè ❤➭♠ t➵✐ x0 ✳ f (x) ❦❤➯ ✈✐ t➵✐ x0 ∈ (a, b) ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ f (x) ❝ã ➤➵♦ ✻✹ P❤Ð♣ tÝ♥❤ ✈✐ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ❈❤ø♥❣ ♠✐♥❤✳ ✰✮ ◆Õ✉ f ❝ã ➤➵♦ ❤➭♠ t➵✐ ➤✐Ó♠ f (x0 + ❦❤➯ ✈✐ t➵✐ x0 t❛ ❝ã x) − f (x0 ) = f (x0 ) x + o( x) A = f (x0 )✳ ✰✮ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ f ❦❤➯ ✈✐ t➵✐ x0 ✱ t❛ ❝ã ❉♦ ➤ã f x0 ✈í✐ f (x0 ) = f (x0 + ❞♦ ➤ã lim f (x0 ) x) − f (x0 ) = A x + o( x) x = A, ♥❣❤Ü❛ ❧➭ f (x0 ) = A✳ ❚õ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ t❤✃② ♠è✐ ❧✐➟♥ ❤Ö s❛✉ df (x) = f (x) x ➜➷❝ ❜✐Öt ♥Õ✉ f (x) = x✱ t❛ ❝ã f (x) = ⇒ dx = df (x) = f (x)dx ❤❛② f (x) = ✷✳✷✳✷ x✳ ❉♦ ➤ã df (x) dx ❈➳❝ q✉② t➽❝ ❧✃② ✈✐ ♣❤➞♥✱ tÝ♥❤ ❜✃t ❜✐Õ♥ ❝ñ❛ ✈✐ ♣❤➞♥ ❝✃♣ ✶ ❛✮ ❈➳❝ ❝➠♥❣ t❤ø❝ q✉② t➽❝ ❧✃② ✈✐ ♣❤➞♥✳ ◆❤ê ❝➳❝ q✉② t➽❝ ❧✃② ➤➵♦ ❤➭♠ ✈➭ ❝➳❝ ❝➠♥❣ t❤ø❝ ➤➲ ♥➟✉ tr➟♥ t❛ ❝ã ❝➳❝ ❝➠♥❣ t❤ø❝ ✈➭ q✉② t➽❝ ❧✃② ✈✐ ♣❤➞♥ s❛✉ ✭✶✮ d(f ± g) = df ± dg ✳ ✭✷✮ d(kf ) = kdf ✭✸✮ d(f g) = f dg + gdf f gdf − f dg ✭✹✮ d = ✈í✐ g(x0 ) = 0✳ g g2 ❜✮ ❚Ý♥❤ ❜✃t ❜✐Õ♥ ❝ñ❛ ✈✐ ♣❤➞♥ ❝✃♣ ✶✳ ◆Õ✉ ❤➭♠ sè y = f (x)✱ tr♦♥❣ ➤ã x ❧➭ ❜✐Õ♥ ➤é❝ ❧❐♣✱ ❧➭ ❤➭♠ sè ❦❤➯ ✈✐ t➵✐ ➤✐Ĩ♠ x ♥➭♦ ➤ã t❤× dy = f (x)dx ❳Ðt ✈✐ ♣❤➞♥ dy tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ x ❧➭ ❜✐Õ♥ ♣❤ơ t❤✉é❝ ✈➭♦ t ♥❣❤Ü❛ ❧➭ y = f (ϕ(t)) ✻✺ ✷✳✷ ❱✐ ♣❤➞♥ ●✐➯ sö x = ϕ(t) ❦❤➯ ✈✐✱ t❛ ❝ã dx = ϕ (t)dt✳ ▼➷t ❦❤➳❝ dy = [f (ϕ(t)] = f (x).ϕ (t) dt ❞♦ ➤ã dy = f (x)ϕ (t)dt ❤❛② dy = f (x)dx✳ ◆❤➢ ✈❐②✱ ✈✐ ♣❤➞♥ dy ❦❤➠♥❣ t❤❛② ➤æ✐ ❦❤✐ x ❧➭ ❜✐Õ♥ ➤é❝ ❧❐♣ ❤❛② ♣❤ô t❤✉é❝✳ ❚Ý♥❤ ❝❤✃t ♥➭② ❣ä✐ ❧➭ tÝ♥❤ ❜✃t ❜✐Õ♥ ❝ñ❛ ✈✐ ♣❤➞♥ ❝✃♣ ✶✳ ị ý trị tr ì ị ĩ ✷✳✷✳✸✳ ❈❤♦ ❤➭♠ sè f (x) ➤➵t δ > s❛♦ ❝❤♦ r➺♥❣ f (x) (a, b)✳ x = c ∈ (a, b) ♥Õ✉ ①➳❝ ➤Þ♥❤ tr♦♥❣ ❦❤♦➯♥❣ ❝ù❝ ➤➵✐ ✭❤♦➷❝ ❝ù❝ t✐Ó✉✮ tt➵✐ ➤✐Ó♠ ❚❛ ♥ã✐ tå♥ t➵✐ f (x) − f (c) < 0(❤♦➷❝ > 0) ✈í✐ ∀x ∈ (c − δ, c + δ) ∩ (a, b), x = c ❇ỉ ➤Ị ✷✳✷✳✹✳ ✭➜Þ♥❤ ❧ý ❋❡r♠❛t✮✳ ◆Õ✉ ❤➭♠ sè c ∈ (a, b) ✈➭ f ❦❤➯ ✈✐ t➵✐ f : (a, b) −→ R c t❤× f (c) = 0✳ c ❧➭ ➤✐Ĩ♠ ❝ù❝ ➤➵✐ ❝đ❛ f ✳ x = c ♥➟♥ tå♥ t➵✐ f (c)✳ ❚❛ ❝ã ❈❤ø♥❣ ♠✐♥❤✳ ➤➵t ❝ù❝ trÞ t➵✐ ●✐➯ sư ❚❤❡♦ ❣✐➯ t❤✐Õt✱ ❞♦ f ❦❤➯ ✈✐ t➵✐ f (c + h) − f (c) h→0 h f (c) = lim f ➤➵t ❝ù❝ ➤➵✐ t➵✐ c ♥➟♥ ❝ãf (c + h) − f (c) < 0✳ ❉♦ ➤ã ❉♦ ✈í✐ h = ❝ã ❣✐➳ trÞ t✉②Ưt ➤è✐ ➤đ ♥❤á t❛ f (c + h) − f (c) > ❦❤✐ h < 0, h f (c + h) − f (c) > ❦❤✐ h < h ❈❤✉②Ĩ♥ q✉❛ ❣✐í✐ ❤➵♥ ❦❤✐ h → t❛ ❝ã f+ (c) ≤ ✈➭ f− (c) ≥ 0✳ ▼➷t ❦❤➳❝✱ ❞♦ tå♥ t➵✐ f (c) ♥➟♥ f+ (c) = f− (c) = f (c) ❤❛② f (c) = 0✳ ❚➢➡♥❣ tù ♥Õ✉ c ❧➭ ➤✐Ĩ♠ ❝ù❝ ➤➵✐ ❝đ❛ f ✱ ❜ỉ ➤Ị ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ✻✻ P❤Ð♣ tÝ♥❤ ✈✐ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè f (x) ①➳❝ ➤Þ♥❤✱ ❧✐➟♥ tơ❝ tr♦♥❣ ➤♦➵♥ [a, b] ✈➭ ❦❤➯ ✈✐ tr♦♥❣ (a, b) ●✐➯ sö f (a) = f (b)✳ ❑❤✐ ➤ã tå♥ t➵✐ c ∈ (a, b) s❛♦ ❝❤♦ f (c) = 0✳ ❍Ư q✉➯ ✷✳✷✳✺✳ ✭➜Þ♥❤ ❧ý ❘♦❧❧❡✮ ❈❤♦ ❤➭♠ sè ❈❤ø♥❣ ♠✐♥❤✳ ♥❤á ♥❤✃t ◆Õ✉ ❱× f ❧✐➟♥ tô❝ tr➟♥ ➤♦➵♥ [a, b] ♥➟♥ ♥ã ➤➵t ❣✐➳ trÞ ❧í♥ ♥❤✃t ✈➭ m tr➟♥ ➤♦➵♥ ➤ã✳ ❚❛ ❝ã M = m t❤× f (x) =❝♦♥st tr➟♥ [a, b] ❤❛② f (x) = 0, ∀x ∈ (a, b) M > m✳ ❑❤✐ ➤ã✱ ❞♦ f (a) = f (b) ♥➟♥ ❣✐➳ trÞ M ➤✐Ĩ♠ c ∈ (a, b)✳ ❚❤❡♦ ➤Þ♥❤ ❧ý ❋❡r♠❛t t❛ ❝ã f (c) = 0✳ ◆Õ✉ ➜Þ♥❤ ❧Ý ✷✳✷✳✻✳ ✭➜Þ♥❤ ❧ý ▲❛❣r❛♥❣❡✮ ◆Õ✉ ❤➭♠ sè ➤♦➵♥ M f (x) ❤♦➷❝ m ➤➵t t➵✐ ♠ét ①➳❝ ➤Þ♥❤✱ ❧✐➟♥ tô❝ tr➟♥ [a, b] ✈➭ ❦❤➯ ✈✐ tr♦♥❣ ❦❤♦➯♥❣ (a, b) t❤× tå♥ t➵✐ Ýt ♥❤✃t ♠ét ➤✐Ĩ♠ c ∈ (a, b) s❛♦ ❝❤♦ f (b) − f (a) = f (c) b−a ❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt ❤➭♠ sè g(x) = f (x) − f (a) − f (b) − f (a) (x − a) b−a g t❤á❛ ♠➲♥ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ c ∈ (a, b) s❛♦ ❝❤♦ g (c) = 0✳ ❚õ ➤ã s✉② r❛ ❉Ô t❤✃② ➤Þ♥❤ ❧ý ❘♦❧❧❡ ♥➟♥ tå♥ t➵✐ ➤✐Ĩ♠ f (b) f (a) = f (c) ba ị ý ợ ❝❤ø♥❣ ♠✐♥❤✳ ❈❤ó ý✳ ◆Õ✉ ➤➷t x = a, x + f (x + ❤❛② x = b t❤× ❝➠♥❣ t❤ø❝ tr➟♥ ➤➢ỵ❝ ✈✐Õt ❞➵♥❣ x) − f (x) = f (x + θ x) x ✈í✐ < θ < y = f (x + θ x) x ✈í✐ < θ < 1✳ ❈➠♥❣ t❤ø❝ tr➟♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ❝➠♥❣ t❤ø❝ sè ❣✐❛ ❣✐í✐ ♥é✐✳ ❍Ư q✉➯ ✷✳✷✳✼✳ ◆Õ✉ f (x) = ✈í✐ ∀x ∈ (a, b) t❤× f (x) =❝♦♥st tr➟♥ (a, b)✳ ✶✶✼ ✸✳✸ ❚Ý❝❤ ♣❤➞♥ s✉② ré♥❣ +∞ |f (x)|dx f (x) ❧➭ ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥ [a, +∞)✳ ●✐➯ sư ➜Þ♥❤ ❧Ý ✸✳✸✳✸✳ ❈❤♦ a +∞ f (x)dx ❝ị♥❣ ❤é✐ tơ✳ ộ tụ tì a r trờ ợ t ó +∞ f (x)dx ❤é✐ tơ t✉②Ưt ➤è✐✳ ◆Õ✉ a +∞ f (x)dx ❤é✐ tô ♥❤➢♥❣ +∞ +∞ ❜➳♥ ❤é✐ tô✳ f (x)dx a a a ❈❤ø♥❣ ♠✐♥❤✳ |f (x)|dx ♣❤➞♥ ❦ú t❤× t❛ ♥ã✐ ➜➷t b F (b) = f (x)dx, ✈í✐ b>a a ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ tå♥ t➵✐ ❣✐í✐ ❤➵♥ +∞ lim F (b)✳ ❚❤❐t ✈❐②✱ ❧✃② ε > 0✳ b→+∞ b |f (x)|dx ❤é✐ tô✱ t❛ ❝ã |f (x)|dx ❤é✐ tô ❦❤✐ b a a ❝❤✉➮♥ ❈❛✉❝❤②✱ tå♥ t➵✐ M > s❛♦ ❝❤♦ ✈í✐ ∀b1 , b2 > M ✱ ❉♦ → +∞✳ ❚❤❡♦ t✐➟✉ b2 | |f (x)|dx| < ε b1 ❉♦ b2 | b2 |f (x)|dx| = b1 b2 |f (x)|dx ≥ | b1 ♥➟♥ t❛ ❝ã f (x)dx| b1 b2 | f (x)dx| < ε b1 ❉♦ ➤ã✱ ✈í✐ ∀b1 , b2 > M ✱ |F (b1 ) − F (b2 )| < lim F (b) ị ý ợ ứ b→+∞ ❈❤ó ý✳ ❱í✐ ❝➳❝ tÝ❝❤ ♣❤➞♥ ❞➵♥❣ b +∞ f (x)dx, −∞ −∞ f (x)dx t❛ ❝ò♥❣ ❝ã ❝➳❝ ✶✶✽ P❤Ð♣ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ➤Þ♥❤ ❧ý t➢➡♥❣ tù✳ ❱Ý ❞ơ✳ ❳Ðt sù ❤é✐ tơ ❝đ❛ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✳ +∞ e−x2 dx✳ ❚❛ ❝ã ✭✶✮ x2 e−x < , ∀x ∈ [1, +∞) x x ✈➭ t❤❡♦ ❦Õt q✉➯ ✈Ý ❞ơ tr➢í❝✱ +∞ ❤é✐ tơ t❤❡♦ ➤Þ♥❤ ❧ý s♦ s➳♥❤✳ +∞ x2 dx ✭✷✮ ✳ ❚❛ ❝ã (1 − x2 )5 1 dx ❤é✐ tô ♥➟♥ tÝ❝❤ ♣❤➞♥ ➤➲ ❝❤♦ x2 x2 dx ❉♦ +∞ 1 x3 ✭✸✮ dx ❤é✐ tô ♥➟♥ tÝ❝❤ ♣❤➞♥ ➤➲ ❝❤♦ ❤é✐ tô✳ +∞ 1 x2 dx =− ∼ −√ 10 (1 − x ) x x3 + x2 dx✳ ❚❛ ❝ã x3 1 + x2 ∼ x3 x +∞ dx ♣❤➞♥ ❦ú ♥➟♥ tÝ❝❤ ♣❤➞♥ ➤➲ ❝❤♦ ♣❤➞♥ ❦ú✳ x +∞ ln x ✭✹✮ dx✳ ❉ï♥❣ q✉② t➽❝ ▲✬❍♦s♣✐t❛❧ t❛ ❞Ơ ❞➭♥❣ ❝❤ø♥❣ ♠✐♥❤ x2 ➤➢ỵ❝ ln x lim √ = x→+∞ x ❉♦ ln x ❉♦ ➤ã tå♥ t➵✐ M > ➤Ĩ ✈í✐ ∀x ≥ M t❤× √ < 1✳ ❚❛ ❝ã x +∞ M ln x dx = x2 +∞ ln x dx + x2 ln x dx x2 M ✶✶✾ ✸✳✸ ❚Ý❝❤ ♣❤➞♥ s✉② ré♥❣ ❚Ý❝❤ ♣❤➞♥ t❤ø ♥❤✃t ❧➭ tÝ❝❤ ♣❤➞♥ ①➳❝ ➤Þ♥❤✳ ln x ❳Ðt tÝ❝❤ ♣❤➞♥ t❤ø ✷✱ t❛ ❝ã √ < 1✱ ∀x ≥ M ♥➟♥ x ln x < , ∀x ≥ M x x2 ❉♦ tô✳ +∞ M ✸✳✸✳✷ dx ❤é✐ tô ♥➟♥ x2 +∞ ln x x2 M dx ❤é✐ tô✱ ❤❛② tÝ❝❤ ♣❤➞♥ ➤➲ ❝❤♦ ❤é✐ ❚Ý❝❤ ♣❤➞♥ s✉② ré♥❣ ❧♦➵✐ ✷ ●✐➯ sö f (x) ❧➭ ❤➭♠ ❦❤➠♥❣ ❜Þ ❝❤➷♥ tr♦♥❣ ➤♦➵♥ [a, b]✳ ●✐➯ t❤✐Õt f (x) ❜Þ ❝❤➷♥ ✈➭ ❦❤➯ tÝ❝❤ tr♦♥❣ ➤♦➵♥ [a, b − η] ✈í✐ η ∈ (0, b − a) ♥❤➢♥❣ ❦❤➠♥❣ ❦❤➯ tÝ❝❤ tr➟♥ ➤♦➵♥ [b − η, b] ✈➭ lim f (x) = ±∞✳ ➜✐Ĩ♠ b ➤➢ỵ❝ − x→b ❣ä✐ ❧➭ ➤✐Ĩ♠ ❦ú ❞Þ ✭❜✃t t❤➢ê♥❣✮ ❝đ❛ ❤➭♠ sè f (x)✳ ◆Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ ❤÷✉ ❤➵♥ b−η lim f (x)dx η→0 a t❤× ❣✐í✐ ❤➵♥ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ tÝ❝❤ ♣❤➞♥ s✉② ré♥❣ ✭❧♦➵✐ ✷✮ ❝ñ❛ f (x) ❧✃② tr➟♥ [a, b]✳ ❑❤✐ ➤ã t❛ ♥ã✐ b f (x)dx ❤é✐ tô ✈➭ a b−η b f (x)dx = lim f (x)dx η→0 a a ❚r➢ê♥❣ ❤ỵ♣ tÝ❝❤ ♣❤➞♥ ❦❤➠♥❣ ❤é✐ tơ t❛ ♥ã✐ ♥ã ♣❤➞♥ ❦ú✳ ❚➢➡♥❣ tù ♥Õ✉ f (x) ❜Þ ❝❤➷♥ ✈➭ ❦❤➯ tÝ❝❤ tr➟♥ ♠ä✐ ➤♦➵♥ [a + η, b] ♥❤➢♥❣ ❦❤➠♥❣ ❦❤➯ tÝ❝❤ tr➟♥ ♠ä✐ ➤♦➵♥ [a, a + η] ✈➭ lim f (x) = ±∞✱ x→a+ t❛ ➤Þ♥❤ ♥❣❤Ü❛ b b f (x)dx = lim a η→0 a+η f (x)dx ✶✷✵ P❤Ð♣ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ế số rờ ợ f (x) ị t ➤✐Ĩ♠ c ∈ (a, b) t❛ ➤Þ♥❤ ♥❣❤Ü❛ b c f (x)dx = b f (x)dx + a a f (x)dx, c ✈➭ tÝ❝❤ ♣❤➞♥ ✈Õ tr➳✐ ❤é✐ tô ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❝➯ ❤❛✐ tÝ❝❤ ♣❤➞♥ ✈Õ ♣❤➯✐ ❤é✐ tô✳ ❱Ý ❞ô✳ dx dx π √ √ = lim = lim [arcsin(−1 + η)] = ✳ − x2 η→0 −1+η − x2 η→0 −1 1 ✭✷✮ ❳Ðt tÝ♥❤ ❤é✐ tơ ❝đ❛ tÝ❝❤ ♣❤➞♥ t❤❡♦ a a x ✰✮ ◆Õ✉ a ≤ 0✱ ➤➞② ❧➭ tÝ❝❤ ♣❤➞♥ ①➳❝ ➤Þ♥❤ ♥➟♥ râ r➭♥❣ ♥ã ❤é✐ tô ✰✮ ◆Õ✉ a = t❛ ❝ã ✭✶✮ 1 = − lim ln η η→0 xa ●✐í✐ ❤➵♥ tr➟♥ ❦❤➠♥❣ tå♥ t➵✐ ♥➟♥ tÝ❝❤ ♣❤➞♥ ♣❤➞♥ ❦ú✳ ✰✮ ◆Õ✉ a = t❛ ❝ã 1 η 1−a − = lim xa η→0 − a a > 1✱ tÝ❝❤ ♣❤➞♥ ♣❤➞♥ ❦ú < a < 1✱ tÝ❝❤ ♣❤➞♥ ❤é✐ tô✳ ❱❐② tÝ❝❤ ♣❤➞♥ ➤➲ ❝❤♦ ❤é✐ tơ ❦❤✐ a < ➜Þ♥❤ ❧Ý ✸✳✸✳✹✳ ✭❚✐➟✉ ❝❤✉➮♥ s♦ s➳♥❤✮ f (x), g(x) ❧➭ ❤❛✐ ❤➭♠ sè ❦❤➠♥❣ ➞♠✱ ❦❤➯ tÝ❝❤ tr➟♥ (a, b] ❝ã ➤✐Ĩ♠ ❦ú ❞Þ x = a✳ ●✐➯ sư f (x) ≤ g(x), ∀x ∈ (a, b]✳ ❑❤✐ ➤ã ✭✶✮ ❈❤♦ b b g(x)dx ❤é✐ tơ t❤× ✰✮ ◆Õ✉ a b f (x)dx ❤é✐ tô✳ a b f (x)dx ♣❤➞♥ ❦ú t❤× ✰✮ ◆Õ✉ a g(x)dx ♣❤➞♥ ❦ú✳ a ✈➭ ❝ï♥❣ ✶✷✶ ✸✳✸ ❚Ý❝❤ ♣❤➞♥ s✉② ré♥❣ ✭✷✮ ●✐➯ sö ❤❛✐ ❤➭♠ sè ❦❤➠♥❣ ➞♠ f (x) ✈➭ g(x) ❦❤➯ tÝ❝❤ tr➟♥ (a, b]✱ ❝ï♥❣ x = a✳ ✈➭ tå♥ t➵✐ ❣✐í✐ ❤➵♥ ❝ã ➤✐Ĩ♠ ❦ú ❞Þ lim+ x→a f (x) = k, (0 < k < +∞) g(x) b b f (x)dx✱ ❑❤✐ ➤ã tÝ❝❤ ♣❤➞♥ s✉② ré♥❣ a g(x)dx ❝ï♥❣ ❤é✐ tô ❤♦➷❝ ❝ï♥❣ ♣❤➞♥ a ❦ú✳ ❈❤ø♥❣ ♠✐♥❤✳ ◆❤❐♥ ①Ðt r➺♥❣ ✈í✐ f (x) ❧➭ ❤➭♠ sè ❦❤➠♥❣ ➞♠✱ ❦❤➯ tÝ❝❤ tr➟♥ (a, b] b ✈➭ ❝ã ể ỳ ị f (x)dx tì F () ệ ♥❣❤Þ❝❤ x = a✱ ♥Õ✉ ➤➷t F (η) = a+η b ❜✐Õ♥ ♥❣❤Ü❛ ❧➭ t➝♥❣ ❦❤✐ F (η) ❤é✐ tô ❧➭ η ❣✐➯♠ ✈Ị 0✳ ❉♦ ➤ã ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤Ĩ f (x)dx a ❜Þ ❝❤➷♥ tr➟♥✳ ❚õ ➤➞②✱ ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù ➤Þ♥❤ ❧ý ✸✳✸✳✶ t❛ ❝ã ❝➳❝ ❦Õt ❧✉❐♥ ❝đ❛ ➤Þ♥❤ ❧ý tr➟♥✳ ❍Ư q✉➯ ✸✳✸✳✺✳ ❈❤♦ ❤❛✐ ❤➭♠ sè ❞➢➡♥❣ f (x) ✈➭ g(x) ❧➭ ❤❛✐ ❤➭♠ sè ❞➢➡♥❣ ❦❤➯ (a, b]✳ tÝ❝❤ tr➟♥ +∞ +∞ f (x) = ✈➭ ♥Õ✉ g(x)dx ❤é✐ tô t❤× f (x)dx ❤é✐ tơ✳ x→a g(x) a a +∞ +∞ f (x) = +∞ ✈➭ ♥Õ✉ ✰✮ ◆Õ✉ lim g(x)dx ♣❤➞♥ ❦ú t❤× f (x)dx ❝ị♥❣ x→a+ g(x) a a ✰✮ ◆Õ✉ lim+ ♣❤➞♥ ❦ú✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ❤Ư q✉➯ ✸✳✸✳✷✳ ❱Ý ❞ơ✳ ❳Ðt sù ❤é✐ tơ ❝đ❛ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉ dx ✭✶✮ I = √ ✳ ❉Ơ t❤✃② x = ❧➭ ➤✐Ĩ♠ ❦ú ❞Þ✳ ❚❛ ❝ã − x2 √ ❉♦ 1 ∼ √ ❦❤✐ x → √ 4 1−x 1−x d(1 − x) √ ❤é✐ tô ♥➟♥ I ❤é✐ tô 1−x ✶✷✷ ✭✷✮ J = P❤Ð♣ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè +∞ ln(1 + x) dx✳ ❚❛ ✈✐Õt xn +∞ ln(1 + x) dx + xn J= ✰✮ ❳Ðt J1 = ln(1 + x) dx xn ln(1 + x) dx t❛ ❝ã xn ln(1 + x) ln(1 + x) =1 ∼ n−1 ✈× lim n x→0 x x x ❉♦ ➤ã J1 ❤é✐ tô ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ n − < ❤❛② n < +∞ ln(1 + x) dx✳ ✰✮ ❳Ðt J2 = xn +∞ • ◆Õ✉ n ≤ t❛ ❝ã dx ♣❤➞♥ ❦ú✳ ▼➷t ❦❤➳❝ n x ln(1 + x) dx > n dx, ∀x ≥ n x x ◆➟♥ J2 ♣❤➞♥ ❦ú✳ • ◆Õ✉ n > 1✳ ❑❤✐ ➤ã ✈✐Õt n = m + ε, m > 1, ε > 0✳ ln(1 + x) ❚❤❡♦ q✉② t➽❝ ▲✬❍♦s♣✐t❛❧✱ lim = ♥➟♥ tå♥ t➵✐ M > x→+∞ xε ln(1 + x) ➤Ĩ ✈í✐ ∀x ≥ M ✱ t❛ ❝ã ♥➟♥ tÝ❝❤ ♣❤➞♥ ❤é✐ tô✳ +∞ M dx ❤é✐ tô✱ ❞♦ ➤ã xm +∞ ln(1 M + x) dx xn ✸✳✸ ❚Ý❝❤ ♣❤➞♥ s✉② ré♥❣ ✶✷✸ ❉♦ ➤ã J2 ❤é✐ tô ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ n > 1✳ ❱❐② tÝ❝❤ ♣❤➞♥ ➤➲ ❝❤♦ ❤é✐ tô ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ < n < 2✳ ✶✷✹ P❤Ð♣ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ✸✳✹ ❇➭✐ t❐♣ ❝❤➢➡♥❣ ■■■ ❚Ý❝❤ ♣❤➞♥ ❜✃t ➤Þ♥❤✳ ❇➭✐ ✶✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✿ (2x + 1)100 (x − 1)2 dx 3/4 )x dx x2 cos4 xdx (1 − x(arctan x)2 dx xdx √ + x2 cos x + cos5 x 11 dx sin2 x + sin4 x dx 13 √ x + a2 √ 15 x2 x2 + 1dx 17 19 ❇➭✐ ✷✳ 12 14 16 dx − x2 dx x + 4x2 + 18 x5 20 ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✿ xn ln xdx arcsin xdx x2 ❇➭✐ ✸✳ 10 ex + dx ex + e3x + dx ex + dx (x + a)2 (x + b)2 dx (x + 1)(x + 2)2 cos4 x dx sin3 x dx x −1 dx √ x − a2 dx √ (x − 1) − x2 dx x +1 x2 dx (1 − x)100 xn eax dx √ x ln(x + + x2 ) √ + x2 xn lnk xdx dx (x + a2 )2 ❉ï♥❣ ❝➠♥❣ t❤ø❝ tr✉② ❤å✐ ❤➲② tÝ♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✿ 1.In = sinn xdx 2.In = tan2n xdx ✶✷✺ ✸✳✹ ❇➭✐ t❐♣ ❝❤➢➡♥❣ ■■■ ❚Ý♥❤✿ (1 + x)2 √ dx x x dx ❇➭✐ ✹✳ √ x x + 3dx (3x + 1)2 dx √ x x + 8x + √ x2 − 2x + 5dx √ − 2x − x2 dx dx x x2 − 2x + 10 x3 (1 − 5x2 )10 dx √ 12 arctan xdx dx √ 14 √ 1+x− 41+x √ dx (x + 1) − x2 11 (x2 + 1)e−3x dx 13 x5 (2 − 5x3 )2/3 dx √ √ 15 x2 dx x6 + dx √ x+1+ x+1 √ 16 e5x sin2 xdx 17 cos(ln x)dx √ 1+ 4x 19 √ dx x+1 dx √ 21 √ − 2x − − 2x x2 23 √ dx x2 + x + 18 20 22 24 ln(sin x)dx sin x √ x √ dx x+1 3x + √ dx −x2 + 6x − xdx √ + x − x2 ❚Ý❝❤ ♣❤➞♥ ①➳❝ ➤Þ♥❤✳ ❉ï♥❣ ➤Þ♥❤ ♥❣❤Ü❛ tÝ♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ ①➳❝ ➤Þ♥❤ s❛✉✿ ❇➭✐ ✶✳ (3x + 1)dx x3 dx π π cos 2xdx sin xdx π ❚Ý♥❤ e ❇➭✐ ✷✳ | ln x|dx 1/e 2 f (x)dx ✈í✐ f (x) = x 0≤x≤1 2 − x 1 0) n→∞ k=0 an + kb n−1 k lim n→∞ k=0 n2 lim ❇➭✐ ✽✳ lim ❈❤♦ f (x) ❧➭ ❤➭♠ sè ❧✐➟♥ tô❝ tr➟♥ [a, b] ✈➭ f (a + b − x) = f (x)✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ b a+b xf (x)dx = a ➳♣ ❞ô♥❣ tÝ♥❤ I = π b f (x)dx a x cos2 x sin3 xdx✳ ❇➭✐ ✾✳ ❈❤♦ f (x) ❧➭ ❤➭♠ ❝❤➼♥ tr➟♥ [−a, a], (a > 0)✳ ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ a a f (x)dx = bx + −a ❞ô♥❣ tÝ♥❤ I = −π/2 ❇➭✐ ✶✵✳ cos2 xdx ✳ 2x + ❈❤♦ ❤➭♠ sè f (x) ❧✐➟♥ tô❝ tr➟♥ R ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ a2 a x3 f (x2 )dx = xf (x)dx, ❞ô♥❣ tÝ♥❤ I = π x3 sin(x2 )dx ❇➭✐ ✶✶✳ (a > 0) √ ➳♣ (b > 0) π/2 ➳♣ f (x)dx, ❚Ý♥❤ ➤é ❞➭✐ ❝➳❝ ❝✉♥❣ ➤➢ê♥❣ ❝♦♥❣ s❛✉✿ y = ln cos x, ≤ x ≤ a (0 < a < π/2) 1 y = x2 − ln x, (1 ≤ x ≤ e) √ 3 y = x , ( ≤ x ≤ ) 2 x2/3 + y 2/3 = a2/3 , (a > 0) ✶✷✽ P❤Ð♣ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè x = et cos t, y = et sin t, ( ≤ t ≤ 1) ❇➭✐ ✶✷✳ ❚Ý♥❤ ❞✐Ư♥ tÝ❝❤ ♠✐Ị♥ ♣❤➻♥❣ ❣✐í✐ ❤➵♥ ❜ë✐ x2 y 2 + = a b x2 = 2py, y = 2px, (p > 0) π y = x + 1, y = cos x, y = 0, (0 ≤ x ≤ ) y = x2 , x + y = 27 x2 y = , y= ✳ x +9 ❇➭✐ ✶✸✳ ❈❤♦ D ❧➭ ♠✐Ị♥ ❣✐í✐ ❤➵♥ ❜ë✐ y = 2px : ≤ x ≤ 2p ❚Ý♥❤ t❤Ó tÝ❝❤ t tể trò ợ t t q✉❛② q✉❛♥❤ ❖①✳ ✷✳ ❉ q✉❛② q✉❛♥❤ ❖②✳ ❇➭✐ ✶✹✳ ❚Ý♥❤ t❤Ó tÝ❝❤ ✈❐t t❤Ó s✐♥❤ r❛ ❦❤✐ ✶✳ ◗✉❛② ♠✐Ị♥ D ❣✐í✐ ❤➵♥ ❜ë✐ y = 2x − x2 ; y = q✉❛♥❤ ❖①✳ ✷✳ ◗✉❛② ♠✐Ò♥ D ❣✐í✐ ❤➵♥ ❜ë✐ y = sin x; y = 0; ≤ x ≤ π q✉❛♥❤ ❖①✳ ❇➭✐ ✶✺✳ ❚Ý♥❤ t❤Ĩ tÝ❝❤ ❝đ❛ ❊❧✐♣s♦✐❞ x2 y z + + ≤ a2 b c ❇➭✐ ✶✻✳ ❚Ý♥❤ t❤Ĩ tÝ❝❤ ❦❤✐ q✉❛② ❤×♥❤ ❊❧✐♣ E : x2 + ✶✳ ◗✉❛♥❤ ❖① y2 ≤1 ✷✳ ◗✉❛♥❤ ❖②✳ ❚Ý❝❤ ♣❤➞♥ s✉② ré♥❣✳ ❇➭✐ ✶✳ ❚Ý♥❤ ❝➳❝ tÝ❝❤ ♣❤➞♥ s✉② ré♥❣ s❛✉✿ +∞ dx +∞ (α > 1) α x +∞ dx e x ln x +∞ xdx √ x3 + +∞ dx −∞ x + +∞ xdx x +8 e 0 +∞ (1 +∞ −2x e−3x dx cos xdx + 2x)dx + 1) x2 (x ✶✷✾ ✸✳✹ ❇➭✐ t❐♣ ❝❤➢➡♥❣ ■■■ ❇➭✐ ✷✳ b a ❳Ðt sù ❤é✐ tơ ❝đ❛ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉✿ dx (b − x)2 ln xdx √ x dx x e − cos x ❇➭✐ ✸✳ +∞ x3 e−x dx 1 +∞ x +1 √ dx x 1 ex dx x3 +∞ sin xdx √ √x x dx sin x e −1 ❈❤ø♥❣ ♠✐♥❤ r➺♥❣ ❝➳❝ tÝ❝❤ ♣❤➞♥ s❛✉ ❤é✐ tô ♥❤➢♥❣ ❦❤➠♥❣ ❤é✐ tơ t✉②Ưt ➤è✐ ✭❜➳♥ ❤é✐ tơ✮ +∞ sin xdx √ ✶✳ x +∞ cos xdx √ x ✶✸✵ P❤Ð♣ tÝ♥❤ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ❤➭♠ sè ♠ét ❜✐Õ♥ sè ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ◆❣✉②Ơ♥ ➜×♥❤ ❚rÝ✱ ❚➵ ❱➝♥ ➜Ü♥❤✱ ◆❣✉②Ô♥ ❍å ◗✉ú♥❤✳ ❤ä❝ ❝❛♦ ❝✃♣ ✲ ❚❐♣ ✶✳ ◆❤➭ ①✉✃t ❜➯♥ ●✐➳♦ ❉ô❝✱ ✷✵✵✽✳ ❬✷❪ ◆❣✉②Ơ♥ ➜×♥❤ ❚rÝ✱ ❚➵ ❱➝♥ ➜Ü♥❤✱ ◆❣✉②Ơ♥ ❍å ◗✉ú♥❤✳ ❤ä❝ ❝❛♦ ❝✃♣ ✲ ❚❐♣ ✷✳ ❬✹❪ P❤➵♠ ◆❣ä❝ ❚❤❛♦✳ ❚♦➳♥ ◆❤➭ ①✉✃t ❜➯♥ ●✐➳♦ ❉ơ❝✱ ✷✵✵✽✳ ❬✸❪ ◆❣✉②Ơ♥ ➜×♥❤ ❚rÝ✱ ❚➵ ❱➝♥ ➜Ü♥❤✱ ◆❣✉②Ô♥ ❍å ◗✉ú♥❤✳ ❤ä❝ ❝❛♦ ❝✃♣ ✲ ❚❐♣ ✸✳ ❚♦➳♥ ❚♦➳♥ ◆❤➭ ①✉✃t ❜➯♥ ●✐➳♦ ❉ô❝✱ ✷✵✵✽✳ ❇➭✐ ❚❐♣ ●✐➯✐ ❚Ý❝❤ ✲ ❚❐♣ ✶✳ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐✱ ✶✾✾✽✳ ❬✺❪ P❤➵♠ ◆❣ä❝ ❚❤❛♦✳ ❇➭✐ ❚❐♣ ●✐➯✐ ❚Ý❝❤ ✲ ❚❐♣ ✷✳ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐✱ ✶✾✾✽✳ ❬✻❪ ◆❣✉②Ô♥ ❳✉➞♥ ▲✐➟♠✳ ❚❐♣ ✶✳ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐✱ ✷✵✶✵✳ ❬✼❪ ◆❣✉②Ô♥ ❳✉➞♥ ▲✐➟♠✳ ❚❐♣ ✷✳ ●✐➯✐ ❚Ý❝❤ ✲ ●✐➳♦ tr×♥❤ ❧ý t❤✉②Õt ✈➭ ❜➭✐ t❐♣ ✲ ●✐➯✐ ❚Ý❝❤ ✲ ●✐➳♦ tr×♥❤ ❧ý t❤✉②Õt ✈➭ ❜➭✐ t❐♣ ✲ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐✱ ✷✵✶✵✳ ❬✽❪ ◆❣✉②Ô♥ ❱➝♥ ❑❤✉➟✱ ▲➟ ▼❐✉ ❍➯✐✳ ●✐➯✐ tÝ❝❤ ❚♦➳♥ ❤ä❝ ✲ ❚❐♣ ✶✳ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❍➭ ◆é✐✱ ✷✵✵✼✳ ❬✾❪ ◆❣✉②Ô♥ ❱➝♥ ❑❤✉➟✱ ▲➟ ▼❐✉ ❍➯✐✳ ●✐➯✐ tÝ❝❤ ❚♦➳♥ ❤ä❝ ✲ ❚❐♣ ✷✳ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❍➭ ◆é✐✱ ✷✵✵✼✳ ... + ❤❛② f (k +1) (x) = (? ?1) k +1 (k + 1) ! (x − 1) k+2 ❚❤❐t ✈❐②✱ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ f (k +1) (x) = [f (k) (x)] (? ?1) k k! = (x − 1) k +1 (? ?1) k k!(k + 1) = − (x − 1) k+2 (? ?1) k +1 (k + 1) ! = (x − 1) k+2 ❈➳❝ ♣❤Ð♣... x→+∞√xα + x2 lim x→∞ x 11 lim (x − ln3 x) lim x→+∞ 13 lim (cos x )1/ x x→0 15 lim x→∞ ✷✳ ✸✳ ✹✳ 2x 10 lim sin(x − 1) tan πx x? ?1 1 12 lim − x→0 arctan x x 14 lim xx x→0 16 lim (ex + x )1/ x x→0 ❈❤ø♥❣ ♠✐♥❤... ln(x + + x2 ) y = 11 y = ex ln sin x 13 y = x1/x 1+ x 1? ??x 17 y = (2x + 3)(2x−5) 2 e−x arcsin e−x 19 y = √ − 2−2x2 15 y = arctan ❇➭✐ ✸✳ y = y = y = 1 + √ + 3√ x x x m+n (1 − x)m (1 + x)n cosn x √