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Tich-phan-Ly-thuyet-va-bai-tap - Tran S Tung Tch phan Nhac...

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Traàn Só Tuøng Tích phaân Trang 1 Nhaéc laïi Giôùi haïn – Ñaïo haøm – Vi phaân 1. Caùc giôùi haïn ñaëc bieät: a) fi = x 0 sinx lim 1 x Heä quaû: fi = x 0 x lim 1 sinx fi = u(x) 0 sinu(x) lim 1 u(x) fi = u(x) 0 u(x) lim 1 sinu(x) b) x x 1 lim 1 e, x R x fi¥ ° ± + = ˛ ² ³ L l Heä quaû: 1 x x 0 lim(1 x) e. fi + = x 0 ln(1 x) lim 1 x fi + = x x 0 e 1 lim 1 x fi - = 2. Baûng ñaïo haøm caùc haøm soá sô caáp cô baûn vaø caùc heä quaû: (c)’ = 0 (c laø haèng soá) 1 (x )' x a a- = a 1 (u )' u u' a a- = a 2 1 1 ' x x ° ± = - ² ³ L l 2 1 u' ' u u ° ± = - ² ³ L l ( ) 1 x ' 2 x = ( ) u' u ' 2 u = x x (e )' e = u u (e )' u'.e = x x (a )' a .lna = u u (a )' a .lna . u' = 1 (ln x )' x = u' (ln u )' u = a 1 (log x ') x.lna = a u' (log u )' u.lna = (sinx)’ = cosx (sinu)’ = u’.cosu 2 2 1 (tgx)' 1 tg x cos x = = + 2 2 u' (tgu)' (1 tg u).u' cos u = = + 2 2 1 (cotgx)' (1 cotg x) sin x - = = - + 2 2 u' (cotgu)' (1 cotg u).u' sin u - = = - + 3. Vi phaân: Cho haøm soá y = f(x) xaùc ñònh treân khoaûng (a ; b) vaø coù ñaïo haøm taïi x (a; b) ˛ . Cho soá gia D x taïi x sao cho x x (a; b) + D ˛ . Ta goïi tích y’. D x (hoaëc f’(x). D x) laø vi phaân cuûa haøm soá y = f(x) taïi x, kyù hieäu laø dy (hoaëc df(x)). dy = y’. D x (hoaëc df(x) = f’(x). D x AÙp duïng ñònh nghóa treân vaøo haøm soá y = x, thì dx = (x)’ D x = 1. D x = D x Vì vaäy ta coù: dy = y’dx (hoaëc df(x) = f’(x)dx)
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Tích phaân Traàn Só Tuøng Trang 2 NGUYEÂN HAØM VAØ TÍCH PHAÂN 1. Ñònh nghóa: Haøm soá F(x) ñöôïc goïi laø nguyeân haøm cuûa haøm soá f(x) treân khoaûng (a ; b) neáu moïi x thuoäc (a ; b), ta coù: F’(x) = f(x). Neáu thay cho khoaûng (a ; b) laø ñoaïn [a ; b] thì phaûi coù theâm: F'(a ) f(x) vaø F'(b ) f(b) + - = = 2. Ñònh lyù: Neáu F(x) laø moät nguyeân haøm cuûa haøm soá f(x) treân khoaûng (a ; b) thì : a/ Vôùi moïi haèng soá C, F(x) + C cuõng laø moät nguyeân haøm cuûa haøm soá f(x) treân khoaûng ñoù. b/ Ngöôïc laïi, moïi nguyeân haøm cuûa haøm soá f(x) treân khoaûng (a ; b) ñeàu coù theå vieát döôùi daïng: F(x) + C vôùi C laø moät haèng soá. Ngöôøi ta kyù hieäu hoï taát caû caùc nguyeân haøm cuûa haøm soá f(x) laø f(x)dx. ´ Do ñoù vieát: f(x)dx F(x) C = + ´ Boå ñeà : Neáu F ¢ (x) = 0 treân khoaûng (a ; b) thì F(x) khoâng ñoåi treân khoaûng ñoù. 3. Caùc tính chaát cuûa nguyeân haøm: ( ) f(x)dx ' f(x) = ´ af(x)dx a f(x)dx (a 0) = ´ ´ [ ] f(x) g(x) dx f(x)dx g(x)dx + = + ´ ´ ´ [ ] [ ] f(t)dt F(t) C f u(x) u'(x)dx F u(x) C F(u) C (u u(x)) = + µ = + = + = ´ ´ 4. Söï toàn taïi nguyeân haøm: Ñònh lyù : Moïi haøm soá f(x) lieân tuïc treân ñoaïn [a ; b] ñeàu coù nguyeân haøm treân ñoaïn ñoù. §Baøi 1 : NGUYEÂN HAØM
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Traàn Só Tuøng Tích phaân Trang 3 BAÛNG CAÙC NGUYEÂN HAØM Nguyeân haøm cuûa caùc haøm soá sô caáp thöôøng gaëp Nguyeân haøm cuûa caùc haøm soá hôïp (döôùi ñaây u = u(x)) dx x C = + ´ du u C = + ´ 1 x x dx C ( 1) 1 a+ a = + a „ - a + ´ 1 u u du C ( 1) 1 a+ a = + a „ - a + ´ dx ln x C (x 0) x = + ´ du ln u C (u u(x) 0) u = + = ´ x x e dx e C = + ´ u u e du e C = + ´ x x a a dx C (0 a 1) lna = + < ´ u u a a du C (0 a 1) lna = + < ´ cosxdx sinx C = + ´ cosudu sin u C = + ´ sinxdx cosx C = - + ´ sin udu cosu C = - + ´ 2 2 dx (1 tg x)dx tgx
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