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MӨC LӨC PHҪ1ĈҤI SӔ VÀ GIҦI TÍCH ..................................................... 1 Chӫ ÿӅ 1. MӋQKÿӅ , tұp hӧp, hàm sӕ ................................................................................... 2 &KX\rQÿӅ 0ӋQKÿӅWұp hӧp  2 &KX\rQÿӅ +jPVӕ EұFQKҩWKjPVӕ Eұc hai  11 Chӫ ÿӅ 3KѭѫQJWUuQKKӋ SKѭѫQJWUuQKEҩWÿҷQJWKӭFYjEҩWSKѭѫQJWUuQK ............. 24 &KX\rQÿӅ 3KѭѫQJWUuQK – hӋ SKѭѫQJWUuQK  24 &KX\rQÿӅ %ҩWÿҷQJWKӭc – EҩWSKѭѫQJWUuQK  40 Chӫ ÿӅ 3. Hàm sӕ OѭӧQJJLiF– 3KѭѫQJWUuQKOѭӧQJJLiF ................................................ 48 &KX\rQÿӅ &{QJWKӭFOѭӧQJJLiF  48 &KX\rQÿӅ +jPVӕ OѭӧQJJLiF  54 &KX\rQÿӅ 3KѭѫQJWUuQKOѭӧQJJLiF  63 Chӫ ÿӅ 4. Tә hӧp – ;iFVXҩt ............................................................................................... 76 &KX\rQÿӅ 4X\WҳFÿӃP+RiQYӏ &KӍQKKӧS7ә hӧp  76 &KX\rQÿӅ ;iFVXҩt  89 &KX\rQÿӅ 1Kӏ thӭF1HZWRQ  98 Chӫ ÿӅ 5. Dãy sӕ – *LӟLKҥn – Ĉҥo hàm .......................................................................... 108 &KX\rQÿӅ 'm\Vӕ&ҩp sӕ cӝQJ– cҩp sӕ QKkQ  108 &KX\rQÿӅ *Lӟi hҥQGm\Vӕ  122 &KX\rQÿӅ *Lӟi hҥQKjPVӕ+jPVӕ OLrQWөc  129 &KX\rQÿӅ Ĉҥo hàm  141 &KX\rQÿӅ 3KѭѫQJWUuQKWLӃSWX\ӃQ  153 Chӫ ÿӅ 6. ӬQJGөQJÿҥRKjPÿӇ khҧRViWKjPVӕ........................................................... 161 &KX\rQÿӅ 7tQKÿѫQÿLӋu cӫa hàm sӕ  161 V

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&KX\rQÿӅ  &ӵFWUӏ cӫa hàm sӕ  175 &KX\rQÿӅ *LiWUӏ lӟQQKҩWJLiWUӏ QKӓ QKҩt cӫa hàm sӕ  191 &KX\rQÿӅ 7LӋm cұQFӫDÿӗ thӏ hàm sӕ  199 &KX\rQÿӅ Ĉӗ thӏ hàm sӕ  205 Chӫ ÿӅ 7. Hàm sӕ PNJKjPVӕ ORJDULWYjKjPVӕ ONJ\WKӯD.............................................. 219 &KX\rQÿӅ /NJ\WKӯa và logaULW  219 &KX\rQÿӅ +jPVӕ PNJ- hàm sӕ ONJ\WKӯa – hàm sӕ ORJDULW  226 &KX\rQÿӅ 3KѭѫQJWUuQKPNJ– SKѭѫQJWUuQKORJDULW  240 &KX\rQÿӅ %ҩWSKѭѫQJWUuQKPNJ– EҩWSKѭѫQJWUuQKORJDULW  254 Chӫ ÿӅ 1JX\rQKjP– Tích phân ................................................................................. 261 &KX\rQÿӅ 1JX\rQKjP  261 &KX\rQÿӅ 7tFKSKkQ  274 &KX\rQÿӅ ӬQJGөQJWtFKSKkQ  286 Chӫ ÿӅ 9. Sӕ phӭc ............................................................................................................... 295 &KX\rQÿӅ &iFSKpSWRiQWUrQWұp hӧp sӕ phӭc  295 &KX\rQÿӅ 7ұp hӧSÿLӇPELӇXGLӉQVӕ phӭc  305

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CHÓŞ Ĉӄ 8

NGUYĂ&#x160;N HĂ&#x20AC;M, TĂ?CH PHĂ&#x201A;N VĂ&#x20AC; ÓŹNG DÓ¨NG TĂ?CH PHĂ&#x201A;N &+8<Ă&#x2021;1Ä&#x2C6;Ó&#x201E; 1

NGUYĂ&#x160;N HĂ&#x20AC;M A. KIÓ&#x201A;N THÓŹC TRÓ?NG TĂ&#x201A;M 1. NguyĂŞn hĂ m Cho hĂ m sÓ&#x2022; y

f x [iFĂżÓ?nh trĂŞn tŇąp K (K lĂ khoҧQJĂżRŇĽn hay nÓąa khoҧng).

Ä&#x2C6;Ó?QKQJKĆ­D HĂ m sÓ&#x2022; F x ÿѭӧc gÓ&#x2018;i lĂ  nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; y

F c x

Ä&#x2C6;Ó?nh lĂ­ 1

Ä&#x2C6;Ó?nh lĂ­ 2

f x trĂŞn K nÓ&#x192;u

f x vÓ&#x;i mÓ&#x2018;i x thuÓ?c K.

NÓ&#x192;u F x lĂ mÓ?t nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; f x trĂŞn K thĂŹ vÓ&#x;i mÓ&#x203A;i hŇľng sÓ&#x2022; C, hĂ m sÓ&#x2022; G x

F x  C FÇ&#x160;QJOjPÓ?t nguyĂŞn hĂ m cÓŤa f x trĂŞn K.

NÓ&#x192;u F x lĂ mÓ?t nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; f x trĂŞn K thĂŹ mÓ&#x2018;i nguyĂŞn hĂ m cÓŤa f x trĂŞn K ĂżÓ&#x2026;u cĂł dŇĽng F x  C vÓ&#x;i C lĂ  hŇľng sÓ&#x2022;.

Ä&#x2C6;Ó?nh lĂ­ 3

MÓ&#x2018;i hĂ m sÓ&#x2022; f x liĂŞn tÓŠc trĂŞn K ĂżÓ&#x2026;u cĂł nguyĂŞn hĂ m trĂŞn K.

TĂ­nh chŇŠt

Âł f c x dx f x  C YÓ&#x;LC OjKŇľQJVÓ&#x2022; x Âł > f ( x) r g ( x) @dx Âł f ( x)dx r Âł g ( x)dx . x Âł kf ( x)dx k Âł f ( x)dx YÓ&#x;L k z 0 . x NÓ&#x192;u F x FyÿҼo hĂ m thĂŹ Âł d F x F x  C . x

2. Bҧng nguyĂŞn hĂ m 1JX\rQKjPFiFKjPVÓ&#x2022;WKŃ­ÓĄQJJŇťS

Âł 0dx

C

1JX\rQKjPPÓŁUÓ?QJ

1

Âł x  a x  b dx

1 xa ln C a b xb

261

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Âł dx Âł1dx

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1

Âłx

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(D z 1)

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2

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x a

2

1

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ln x  C

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Âł e dx

ex  C

Âłe

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2

dx

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x

ax  C (0  a z 1) ln a

x Âł a dx

1

ax  b

dx

mx  n Âł a dx

2

1 x arctan  C a a

dx ln x  x 2  a 2  C

dx

1  C , a z 0

a ax  b

1 ln ax  b  C , a z 0

a 1 ax  b e  C , (a z 0) a a mx  n  C , 0  a z 1

m ln a

Âł cos xdx

sin x  C

Âł cos(ax  b)dx

1 sin(ax  b)  C , a z 0

a

Âł sin xdx

 cos x  C

Âł sin(ax  b)dx

1  cos(ax  b)  C , a z 0

a

1

Âł cos

2

x

dx

tan x  C

1

1 tan ax  b  C , a z 0

a

Âł cos ax  b dx 2

1 Âł sin 2 xdx  cot x  C

Âł sin ax  b dx

1

1 co t ax  b  C , a z 0

a

Âł tan xdx

 ln cos x  C

Âł tan ax  b dx

1  ln cos ax  b  C , a z 0

a

Âł co t xdx

ln sin x  C

Âł co t ax  b dx

1 ln sin ax  b  C , a z 0

a

2

B. CĂ C DҤNG BĂ&#x20AC;I TŇ°P DŇĽQJ1JX\rQKjPFŃŤEҧn 3KŃ­ŃŤQJSKiSJLҧL Ă p dÓŠng cĂĄc cĂ´ng thÓ­c trong bҧQJQJX\rQKjPFŃŤEҧn 2. VĂ­ dÓŠ mŇŻu

262

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GOODA.VN - THU VI áť&#x2020;N SĂ CH QUĂ? VĂ­ dÓŠ 1: Cho hĂ m sÓ&#x2022; f x

2 x3 3   C. 3 2x LÓĄi giҧi chi tiÓ&#x192;t

A.

CĂĄch 1:

Âł

2 x4  3 . TĂ­nh nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; f x . x2

B.

2 x3 3   C. 3 x

2 x4  3 Âł x 2 dx

f x dx

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C. 2 x 3 

§ 2 3¡ ³ ¨Š 2 x  x 2 ¸š dx

3  C. x

D.

2 x3 3   C. 3 x

2 x3 3  C. 3 x

CĂĄch 2: SÓą dÓŠQJPi\WtQK&DVLR

2 vĂ o hĂ m sÓ&#x2022; f x

%˹ͣc 1: Thay x

%˹ͣc 2: Sӹ dӊng SHIFT

35 2 x4  3 WDÿѭӧc kÓ&#x192;t quҧ lĂ 2 4 x

8,75 .

d , x , . dx

Ä&#x2C6;iSiQ$1KŇąp

d § 2X 3 3 ¡  ¨ ¸ x 2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 7,625 , loŇĽLĂżiSiQ$ dx Š 3 2X š

Ä&#x2C6;iSiQ%1KŇąp

d § 2X 3 3 ¡  ¸ x 2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 7, 25 , loŇĽLĂżiSiQ% ¨ dx Š 3 Xš

Ä&#x2C6;iSiQ&1KŇąp

d § 3¡ 3 ¨ 2X  ¸ x dx Š Xš

Ä&#x2C6;iSiQ'1KŇąp

d § 2X 3 3 ¡  ¸ x 2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 8,75 , chÓ&#x2018;QĂżiSiQ' ¨ dx Š 3 Xš

2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 24,75 , loŇĽLĂżiSiQ&

ChÓ&#x2018;n D. VĂ­ dÓŠ 2: HĂ m sÓ&#x2022; f x

x  2

x3

4 2   6. x x2 LÓĄi giҧi chi tiÓ&#x192;t

x  2

x

2

3

'RĂży F x

cĂł nguyĂŞn hĂ m lĂ  hĂ m F x %LÓ&#x192;W F 1 6 NKLĂży F x FyGŇĽQJ

B. ln x 

A. ln x 

f x

2

x2  4 x  4 x3

Âł f x dx Âł

4 2   4. x x2

VĂ­ dÓŠ 3: Cho F x

ax

2

dx dx dx  4Âł 2  4Âł 3 x x x

3035 . LÓĄi giҧi chi tiÓ&#x192;t

ln x 

B. T

D. ln x 

4 2   6. x x2

2018 x

 3 x  1 e 2 x

4 2   C. x x2

4 2   12 . x x2

 bx  c e 2 x lĂ mÓ?t nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; f x

trên khoҧng f; f . Tính T A. T

4 2   12. x x2

1 4 4  2  3. x x x

MĂ F 1 6 Â&#x; C 12 GRĂży F x ln x  ChÓ&#x2018;n C.

C. ln x 

2

a  2b  4c . 1007 .

C. T

5053 .

D. T

1011 .

263

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? VĂŹ F x

ax

2

 bx  c e 2 x lĂ mÓ?t nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; f x

khoҧng f; f QrQWDFy F x c

2018 x

2

 3 x  1 e 2 x trĂŞn

f x , vÓ&#x;i mÓ&#x2018;i x Â? f; f .

Hay 2ax 2  x 2b  2a  2c  b e 2 x

2018 x

2

 3 x  1 e 2 x , vÓ&#x;i mÓ&#x2018;i x Â? f; f .

­ °a 1009 ­2a 2018 ° 2021 ° ° . Ä&#x2C6;Ó&#x2014;ng nhŇŠt hÓ&#x2039; sÓ&#x2022; tDÿѭӧF ÂŽ2b  2a 3 Â&#x153; ÂŽb  2 °2c  b 1 ° ÂŻ 2023 ° °Ě&#x201E;c  4

a  2b  4c

VŇąy T

§ 2021 ¡ § 2023 ¡ 1009  2. ¨  ¸  4. ¨  ¸ 2 š 4 š Š Š

3035 .

ChÓ&#x2018;n A. VĂ­ dÓŠ 4: TĂŹm mÓ?t nguyĂŞn hĂ m F x cÓŤa hĂ m sÓ&#x2022; f x ax 

F 1 4 ; f 1 0 .

b x2

x z 0

3x 2 3 7   . 4 2x 4

B. F x

3x 2 3 7   . 4 2x 4

3x 2 3 7   . 2 4x 4 LÓĄi giҧi chi tiÓ&#x192;t

D. F x

3x 2 3 1   . 2 2x 2

A. F x

C. F x

F x

Âł

f x dx

Âł ax 

b dx x2

biÓ&#x192;t rŇľng F 1 1 ;

ax 2 b  C. 2 x

3 ­a ­ °2  b  C 1 °a 2 ­ F 1 1 ° ° ° 3 °a ° Theo giҧ thiÓ&#x192;t ta cĂł hÓ&#x2039; ÂŽ F 1 4 Â&#x153; ÂŽ  b  C 4 Â&#x153; ÂŽb  . 2 2 ° ° ° f 1 0

ÂŻ 7 °a  b 0 ° ° °C 4 ÂŻ ÂŻ VŇąy hĂ m sÓ&#x2022; f x cĂł mÓ?t nguyĂŞn hĂ m lĂ  F x

3x 2 3 7   . 4 2x 4

ChÓ&#x2018;n A. VĂ­ dÓŠ 5: GÓ&#x2018;i F x lĂ mÓ?t nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; f x

thÓ­c T A. T

2 x , thÓ&#x201C;a mĂŁn F 0

F 0  F 1  F 2  ...  F 2017 . 1009.

22017  1 . ln 2

B. T

22017.2018 .

C. T

22017  1 . ln 2

LÓĄi giҧi chi tiÓ&#x192;t

264

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1 . TĂ­nh giĂĄ trÓ? biÓ&#x2021;u ln 2

D. T

22018  1 . ln 2


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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? 7DFy F x

Âł

f x dx

2x C . ln 2

x Âł 2 dx

MĂ F 0

1 1 Â&#x; C ln 2 ln 2

.KLĂży T

F 0  F 1  F 2  ...  F 2017

1 Â&#x;C ln 2

0 Â&#x; F x

2x . ln 2

20 2 22 22017    ...  ln 2 ln 2 ln 2 ln 2 1 1  22018 . ln 2 1  2

22018  1 . ln 2

ChÓ&#x2018;n D.

DŇĽng 2: TĂŹm QJX\rQKjPEŇľQJSKŃ­ŃŤQJSKiSĂżÓ&#x2122;LELÓ&#x192;n sÓ&#x2022; 1. 3KŃ­ŃŤQJSKiSJLҧL

Âł f u du

NÓ&#x192;u

u x Fy ÿҼo hà m liên tӊc thÏ

F u  C vĂ u

³ f ªu x ºŸ .uc x dx

F u  C.

Chú ý: Công thӭc vi phân

f c x dx d f x

sin xdx d  cos x  d cos x

dx d x  1 d x  2 ... d x  n

cos xdx d sin x

dx

1 d ax

a

1 d ax  1 ... a

1 d ax  n

a

1 dx d ln x , e x dx d e x

x

2. VĂ­ dÓŠ mŇŻu

1

VĂ­ dÓŠ 1: NguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; A.

1  C. 2  4x

2 x  1

1

B.

2 x  1

3

2

C.

 C.

1  C. 4x  2

D.

1  C. 2 x  1

LÓĄi giҧi chi tiÓ&#x192;t

1

Âł 2 x  1

CĂĄch 1: I

2

dx Ä&#x2C6;Ó&#x2122;i biÓ&#x192;n 2 x  1 t Â&#x; dt

dt

Âł 2t

2dx 'RĂży I

2

1 C . 2t

1 C . 2  4x

TrÓŁ lŇĽLSKpSĂżÓ&#x2122;i biÓ&#x192;QWDÿѭӧF I CĂĄch 2: SÓą dÓŠQJYLSKkQ

I

1

Âł 2 x  1

2

dx

1

Âł 2 x  1

2

§ 2x  1 ¡ d¨ ¸ Š 2 š

1 1 d 2 x  1

Âł 2 2 x  1 2

1 1 . C 2 2x 1

1  C. 2  4x

ChÓ&#x2018;n A. VĂ­ dÓŠ 2: TĂŹm nguyĂŞn hĂ m sau I A.

3

3

x 1 4

4

 C.

B.

4

Âł

3

3

x 1

x  1dx .

3

4

 C.

C. 

2

3

x 1 3

2

 C.

D. 

3

3

x 1 2

2

 C. 265

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? LÓĄi giҧi chi tiÓ&#x192;t Ä&#x2C6;Ňťt t I

3

x  1 Â&#x; t3

2 Âł t.3t dt

x  1 Â&#x; 3t 2 dt

3 Âł 3t dt

dx.

3t 4 C. 4

3

TrÓŁ lŇĽLSKpSĂżÓ&#x2122;i biÓ&#x192;QWDÿѭӧF I

3

x 1

4

4

 C.

ChÓ&#x2018;n A.

1  x5

Âł x 1  x dx

VĂ­ dÓŠ 3: NguyĂŞn hĂ m cÓŤa hĂ m I

S 10a  b bŇľng A. 1. LÓĄi giҧi chi tiÓ&#x192;t

1

C. 0.

1  x5  2 x5 Âł x 1  x5 dx

I1

Âł x dx

I2

2 x 4 5 4 Âł x5  1dx Ä&#x2C6;Ňťt x  1 t Â&#x; 5 x dx

I2

Âł 5t dt

§ 1 2x4 ¡ ³ ¨Š x  1  x5 ¸š dx

D. 3. 1 2x4 dx  Âł x Âł x5  1 dx

I1  I 2 .

ln x  C .

2

2 ln t  C 5

ln x 

VŇąy I

cĂł dŇĽng a ln x 5  b ln 1  x 5  C  .KL Ăży WÓ&#x2122;ng

B. 2.

1  x5 Âł x 1  x5 dx

CĂĄch 1: I

5

dt Â&#x; x 4 dx

dt . 5

2 ln x 5  1  C . 5

2 ln x 5  1  C 5

1 ln x 5  2ln 1  x 5  C . 5

1 ; b 2 Â&#x; 10a  b 0. 5 CĂĄch 2: SÍľ dÍ­ng vi phân

'RĂży a

1  x x dx Âł x 1  x

5

I

5

4

5

5 5 1 1  x d x

5 Âł x 5 1  x 5

1 § 1 2 ¡ d x5

¨ 5 5 ¸ ³ 5 Š x 1 x š 6X\UD a

5 5 5 ¡ 1 x  1  2 x d x 1 § x5  1 2 x5 ¨ ¸ d x5

 ³ ³ 5 5 5 5 5 5 5 5 ¨ x 1  x x 1  x ¸ x 1  x

Š š

1 ln x5  2ln 1  x5  C . 5

1 ; b 2 Â&#x; 10a  b 0. 5

ChÓ&#x2018;n C.

DŇĽng 3: NguyĂŞn hĂ m tÓŻQJSKŇŤn 3KŃ­ŃŤQJSKiSJLҧL NÓ&#x192;u u vĂ  v lĂ  hai hĂ m sÓ&#x2022; FyÿҼo hĂ m liĂŞn tÓŠc trĂŞn K thĂŹ Âł udv uv  Âł vdu . 266

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? &iFKËąXWLrQÿ̿WX

nhŇŠt log ln x QKuĂżD ax  b;a x 2  bx  c WDPOŃ­Ó§ng sin x;cos x , tÓ­ PÇ&#x160; e x .a x .

VĂ­ dÓŠ Ä&#x2C6;Ňťt

Âł ln xdx

Âł x ln xdx

Âł x sin xdx

Âł xe dx

Âł e .cos xdx

u vdv

ln x dx

ln x xdx

x

x

cos x

sin xdx

e x dx

e x dx

x

x

2. VĂ­ dÓŠ mŇŻu VĂ­ dÓŠ 1: TĂŹm nguyĂŞn hĂ m sau Âł ln 4xdx .

x ln 4 x  1  C. 4 LÓĄi giҧi chi tiÓ&#x192;t

x ln 4 x  1  C. 2

B.

A.

C. x ln 4 x  1  C.

D. 2 x ln 4 x  1  C.

dx ­ ­u ln 4 x °du Â&#x; CĂĄch 1: Ä&#x2C6;Ňťt ÂŽ x . ÂŽ ÂŻdv dx °¯v x .KLĂży Âł ln 4 xdx uv  Âł vdu

1 x.ln 4 x  Âł x. dx x

x.ln 4 x  Âł 1dx

x ln 4 x  1  C .

CĂĄch 2: SÓą dÓŠQJYLSKkQ

Âł ln 4 xdx

x.ln 4 x  Âł xd ln 4 x

x.ln 4 x  Âł x.

x ln 4 x  1  C.

x ln 4 x  x  C

4 dx 4x

x.ln 4 x  Âł 1dx

Cåch 3: Sӹ dӊQJPi\WtQK&DVLR %˹ͣc 1: Thay x

2 vĂ o hĂ m sÓ&#x2022; f x ln 4 x WDÿѭӧc kÓ&#x192;t quҧ lĂ  2,07944 .

%˹ͣc 2: SÓą dÓŠng SHIFT Ä&#x2C6;iSiQ$1KŇąp

d , x , . dx

d §X ¡ ¨ ln 4X  1 ¸ x dx Š 4 š

2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 0,519 , khĂĄc kÓ&#x192;t quҧ cÓŤDEŃ­Ó&#x;c 1, loŇĽi

d §X ¡ ¨ ln 4X  1 ¸ x dx Š 2 š

2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 1,039 , khĂĄc kÓ&#x192;t quҧ cÓŤDEŃ­Ó&#x;c 1, loŇĽi

d X ln 4X  1 x dx

2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 2,07944 , bŇľng kÓ&#x192;t quҧ cÓŤDEŃ­Ó&#x;c 1

ĂżiSiQ$ Ä&#x2C6;iSiQ%1KŇąp ĂżiSiQ% Ä&#x2C6;iSiQ&1KŇąp chÓ&#x2018;QĂżiSiQ& ChÓ&#x2018;n C.

VĂ­ dÓŠ 2: Cho F x lĂ nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; y A.

1 2x 3 e x  2  . 2 4

B.

1 2x § 1¡ e ¨ x  ¸. 2 Š 2š

1 x.e 2 x , biÓ&#x192;t F 0  . TĂŹm F x . 4 C.

1 2x § 1¡ 5 e ¨x ¸ . 2 Š 2š 4

1¡ 3 § D. 2e 2 x ¨ x  ¸  . 2š 4 Š 267

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? LÓĄi giҧi chi tiÓ&#x192;t

­du dx ­u x ° Âł x.e dx Ä&#x2C6;Ňťt ÂŽÂŻdv e2 x dx Â&#x; ÂŽv 1 e2 x . °Ě&#x201E; 2

CĂĄch 1: I

I

2x

1 2x 1 xe  Âł e 2 x dx 2 2

uv  Âł vdu

F 0

1 0§ 1¡ .e ¨ 0  ¸  C 2 Š 2š

1 C 4

1 2x 1 2x xe  e  C 2 4 1 Â&#x;C 4

1 2x § 1¡ e ¨ x  ¸  C. 2 Š 2š

0 'RĂży F x

1 2x § 1¡ e ¨x ¸. 2 Š 2š

CĂĄch 2: SÓą dÓŠQJYLSKkQ I

§ e 2 x ¡ x.e 2 x e2 x xd  ¨ ¸ ³ Š 2 š 2 ³ 2 dx 1 2x 1 2x 1 2x § 1¡ xe  e  C e ¨ x  ¸  C. 2 4 2 Š 2š

1 2x 1 xe  Âł e 2 x dx 2 2

2x Âł xe dx

ChÓ&#x2018;n B.

DŇĽng 4: NguyĂŞn hĂ m cÓŤa mÓ?t sÓ&#x2022; hĂ m sÓ&#x2022; WKŃ­ÓĄng gŇťS 3KŃ­ŃŤQJSKiSJLҧL 1.1. NguyĂŞn hĂ m hĂ m hÎ&#x2021;u tÍĄ 3KŃ­ŃŤQJSKiSJLҧi: f x lĂ  hĂ m hÓłu tÓ? f x

P x

Q x

NÓ&#x192;u bŇąc cÓŤa P x lÓ&#x;QKŃŤQEŇąc cÓŤa Q x thĂŹ ta thÓľc hiÓ&#x2039;QSKpSFKLDĂżDWKÓ­c. NÓ&#x192;u bŇąc cÓŤa P x lÓ&#x;QKŃŤQEŇąc cÓŤa Q x vĂ Q x cĂł dŇĽng tĂ­ch nhiÓ&#x2026;u nhân tÓą thĂŹ ta phân tĂ­ch f x

thĂ nh tÓ&#x2122;ng cÓŤa nhiÓ&#x2026;u phân thÓ­c.

P x

A B  . xa xb

( x  a )( x  b)

P x

2

( x  m)(ax  bx  c) P x

( x  a )( x  b)

2

A Bx  C .  2 x  m ax  bx  c

A B C .   x  a x  b ( x  b) 2

CĂĄch tĂŹm A, B, C. CĂĄch 1: 4X\ĂżÓ&#x2014;QJĂżÓ&#x2014;QJQKŇŠWKÓ&#x2039;VÓ&#x2022; CĂĄch 2: 7KD\EDJLiWUÓ?EŇŠWNuFÓŤD x, OŇąSUDKÓ&#x2039;EDSKŃ­ŃŤQJWUuQKEDŇ­Q ChĂş Ă˝: CĂĄc cĂ´ng thÓ­c tĂ­nh nhanh cÓŤa nguyĂŞn hĂ m hÓłu tÓ? 1 1 dx ln ax  b  C . 1. Âł ax  b a 1 1 2. Âł dx C. 2 a ax  b

ax  b

268

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? 1

1 xa ln C a b xb

3.

Âł x  a x  b dx

4.

Âłx

5.

Âł x  a x  b x  c dx

2

1 dx  a2

1 ln x  a  ln x  b  C . a b

1 x arctan  C . a a

P x

A.ln x  a  B.ln x  b  C.ln x  c  C0 .

7uP$%&EĚšQJFiFKVÍľ dÍ­QJPi\WtQK&$6,2I[913/86 A

B

C

P x

d X  a X  b X  c x dx

a

P x

d X  a X  b X  c x b dx P x

d X  a X  b X  c x c dx

&$/& a ÿѭӧFNÓ&#x192;WTXҧFKtQKOjJLiWUÓ?FÓŤD$

&$/& b

ÿѭӧFNÓ&#x192;WTXҧFKtQKOjJLiWUÓ?FÓŤD%

&$/& c

ÿѭӧFNÓ&#x192;WTXҧFKtQKOjJLiWUÓ?FÓŤD&

1.2. NguyĂŞn hĂ m hĂ m vĂ´ tÍĄ 3KŃ­ŃŤQJSKiSJLҧi6Óą dÓŠQJSKŃ­ŃŤQJSKiSĂżÓ&#x2122;i biÓ&#x192;n sÓ&#x2022;, sÓą dÓŠng vi phân. MÓ?t sÓ&#x2022; dŇĽQJĂżÓ&#x2122;i biÓ&#x192;n sÓ&#x2022; ÿҝc biÓ&#x2039;W I [ cĂł chÓ­D

&iFKĂżÓ&#x2122;i biÓ&#x192;n

a2  x2

x

a sin t ,



a2  x2

x

a tan t ,



x2  a2

x

1 § ¡ R¨ ¸ Š ( x  a )( x  b) š

a , cos t

2

S 2

dt d t 

t Â? > 0; S @ \

t

§ ax  b ¡ R ¨ x, m ¸ cx  d š Š

S

S 2

S 2

^S ` 2

xa  xb

t

m

ax  b cx  d

1JX\rQKjPKjPOĚ Íťng giĂĄc 3KŃ­ŃŤQJSKiSJLҧL7DVÓąGÓŠQJFiFSKpSELÓ&#x192;QĂżÓ&#x2122;LĂżÓ&#x2021;ÿѭDYÓ&#x2026;QJX\rQKjPKjPFŃŤEҧQ +jPVÓ&#x2022;FyGŇĽQJ f x sin m x.cos n x

sin x. Ć&#x201D;1Ó&#x192;Xm OŇżn FKŇšQÿҝW t cos x Ć&#x201D;1Ó&#x192;Xm OŇżn OŇżÿҝW t sinx KRŇťF t

Ć&#x201D; 1Ó&#x192;Xm FKŇšQn OŇżÿҝW t

cosx .

269

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? Ć&#x201D;1Ó&#x192;Xm FKŇšQn FKŇšQÿҝW t

tan x

&iFF{QJWKÓ­FOŃ­Ó§QJJLiFKD\GQJ

1  cos 2 x 1  cos 2 x , cos 2 x , tan 2 x 2 2

sin 2 x

1 1 . cos 2 x

2. VĂ­ dÓŠ mŇŻu

Âł 2x

9tGÓŠCho I

2

dx b a.ln x  1  ln 2 x  1  C . TĂ­nh giĂĄ trÓ? cÓŤa P  x 1 2

A. 1.

3 . 2

B.

C. 3.

2a  b . D. 0.

LÓĄi giҧi chi tiÓ&#x192;t

Âł 2x

CĂĄch 1: I

2

dx  x 1

1§ 1

2

¡

³ 3 ¨Š x  1  2 x  1 ¸š dx .KLÿy a

1 2 , b  nĂŞn P 3 3

1 1 ln x  1  ln 2 x  1  C . 3 3

2a  b 0 .

CĂĄch 2: SÓą dÓŠng cĂ´ng thÓ­FWtQKQKDQK Âł I

Âł 2x

2

dx  x 1

1 2 x  1  2 x  1

dx 3 Âł x  1 2 x  1

dx

Âł x  1 2 x  1

dx

Âł x  1 2 x  1

1 1 § 1 . ln x  1  ln x  ¨ 2 1 1 Š 2 2

1 1 2x  1 ln x  1  ln C 3 3 2

1 2Âł

¡ ¸C š

1

x  a x  b

1 ln x  a  ln x  b  C. a b

dx

dx 1 x  1 §¨ x  ¡¸ 2š Š

1 2Âł

dx 1 x  1 §¨ x  ¡¸ 2 š Š

1 1 1 ln x  1  ln x   C 3 3 2

1 1 ln x  1  ln 2 x  1  C1 . 3 3

1 2 , b  nĂŞn P 2a  b 0 . 3 3 CĂĄch 3: SÓą dÓŠQJPi\WtQK&DVLR .KLĂży a

a b 2

1 &$/& 1 ÿѭӧFNÓ&#x192;WTXҧ a d 2 2 x  x  1 x 1 dx 1

d 2 x 2  x  1 x dx

.KLĂży a

1 2

1 2 , b  nĂŞn P 3 3

&$/&

1 2

ÿѭӧFNÓ&#x192;WTXҧ

1 . 3 b 2

1  Â&#x;b 3

2 . 3

2a  b 0 .

ChÓ&#x2018;n D.

270

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ?

2 x  1  4ln

A. I

2 x  1  4ln

C. I

dx . 2x 1  4

Âł

VĂ­ dÓŠ 2: TĂŹm nguyĂŞn hĂ m I

2 x  1  4  C.

B. I

2 x  1  4  C.

D. I

2 x  1  4ln

2 x  1  4ln

2 x  1  4 .

2x 1  4 .

LÓĄi giҧi chi tiÓ&#x192;t CĂĄch 1: Ä&#x2C6;Ňťt t

I

2x 1 Â&#x; t 2

tdt

Âłt4 Âł

t 44 dt t4

2 x  1 Â&#x; tdt

dx .

4 ¡

§

³ ¨Š1  t  4 ¸š dt

t  4ln t  4  C Â&#x; I

2 x  1  4ln

2 x  1  4  C.

Cåch 2: Sӹ dӊng måy tính &DVLR %˹ͣc 1: Thay x

1 WDÿѭӧc kÓ&#x192;t quҧ lĂ 0,174457 . 2x 1  4

2 vĂ o hĂ m sÓ&#x2022; f x

%˹ͣc 2: Sӹ dӊng SHIFT

d , x , . dx

d 2 X  1  4ln 2 X  1  4 x 2 WDÿѭӧc kÓ&#x192;t quҧ lĂ 0,174457 , bŇľng kÓ&#x192;t quҧ dx cÓŤDEŃ­Ó&#x;FGRĂżyFKÓ&#x2018;QĂżiSiQ$ NÓ&#x192;u kÓ&#x192;t quҧ khĂ´ng bŇľng kÓ&#x192;t quҧ EŃ­Ó&#x;c 1, ta thÓą tiÓ&#x192;SFiFĂżiSiQNKiFQKŃ­FiFKWUrQ ChÓ&#x2018;n A. Ä&#x2C6;iSiQ$1KŇąp

VĂ­ dÓŠ 3: TĂŹm nguyĂŞn hĂ m I A. sin 3 x  C.

Âł sin

2

x.cosxdx .

B. sin 2 x  C.

C.

LÓĄi giҧi chi tiÓ&#x192;t CĂĄch 1: SÓą dÓŠQJSKŃ­ŃŤQJSKiSĂżÓ&#x2122;i biÓ&#x192;n Ä&#x2C6;Ňťt sin x t Â&#x; cos xdx

dt , ta cĂł I

Âł t dt 2

CĂĄch 2: SÓą dÓŠQJYLSKkQ 2 Âł sin x.cosxdx

2 Âł sin xd sinx

ChÓ&#x2018;n C.

sin 3 x  C. 3

D.

cos 2 x  C. 2

t3  C . TrÓŁ lŇĽLSKpSĂżÓ&#x2122;i biÓ&#x192;n WDÿѭӧc I 3

sin 3 x  C. 3

sin 3 x  C. 3

BĂ&#x20AC;I TŇ°P TÓ&#x2DC;NG HÓŚ3&+8<Ă&#x2021;1Ä&#x2C6;Ó&#x201E; 1

&kX 1: Trong cĂĄc hĂ m sÓ&#x2022; sau, hĂ m sÓ&#x2022; nĂ o cĂł mÓ?t nguyĂŞn hĂ m lĂ  hĂ m sÓ&#x2022; F x ln x ? A. f x

x.

B. f x

1 . x

C. f x

x3 . 2

D. f x

x.

&kX 2: Cho f x , g x lĂ cĂĄc hĂ m sÓ&#x2022; [iFĂżÓ?nh vĂ  liĂŞn tÓŠc trĂŞn \ . Trong cĂĄc mÓ&#x2039;QKĂżÓ&#x2026; sau, mÓ&#x2039;QKĂżÓ&#x2026; nĂ o VDL?

³ f x g x dx ³ f x dx.³ g x dx . C. ³ ª f x  g x Ÿº dx ³ f x dx  ³ g x dx . A.

B. Âł 2 f x dx D.

2 Âł f x dx .

³ ª f x  g x Ÿº dx ³ f x dx  ³ g x dx . 271

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? &kX 3: NÓ&#x192;u

Âł f x dx

1  ln x  C thĂŹ f x lĂ x

A. f x

x  ln x  C .

B. f x  x 

C. f x 

1  ln x  C . x2

D. f x

1  ln x  C . x

x 1 . x2

&kX 4: HĂ m sÓ&#x2022; F x e x lĂ  mÓ?t nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; 3

A. f x e .

B. f x 3 x .e .

x3

2

x3

C. f x

3

ex . 3x 2

D. f x

3

x3 .e x 1 .

&kX 5: TĂŹm nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022; f x e5 x . A.

Âł f x dx

e5 x ln 5+ C .

B.

Âł f x dx

1 5x e +C . 5

C.

Âł f x dx

5e5 x + C .

D.

Âł f x dx

e5x + C .

&kX 6: %LÓ&#x192;t F x lĂ mÓ?t nguyĂŞn hĂ m cÓŤa cÓŤa hĂ m sÓ&#x2022; f x sin x YjĂżÓ&#x2014; thÓ? hĂ m sÓ&#x2022; y

F x ĂżLTXD

§S ¡ ĂżLÓ&#x2021;m M 0;1 . TĂ­nh F ¨ ¸ . Š2š §S ¡ B. F ¨ ¸ 1 . Š2š

§S ¡ A. F ¨ ¸ 2 . Š2š &kX 7: Cho biÓ&#x192;t

2 x  13

Âł x  1 x  2 dx

A. a  2b 8 . &kX 8: HĂ m sÓ&#x2022; y

§S ¡ C. F ¨ ¸ 0 . Š2š

§S ¡ D. F ¨ ¸ 1 . Š2š

a ln x  1  b ln x  2  C . MÓ&#x2039;QKĂżÓ&#x2026; QjRVDXĂżk\Ăż~QJ"

B. a  b 8 .

C. 2a  b 8 .

D. a  b 8 .

f ( x) cĂł mÓ?t nguyĂŞn hĂ m lĂ  F x e 2 x . TĂŹm nguyĂŞn hĂ m cÓŤa hĂ m sÓ&#x2022;

f ( x)  1 . ex

f ( x)  1 f ( x)  1 dx e x  e  x  C . dx 2e x  e  x  C . B. ³ x x e e f ( x)  1 f ( x)  1 1 x x dx e e C . dx 2e x  e  x  C . C. ³ D. ³ x x e 2 e x3 &kX 9: Khi tính nguyên hà m ³ x  1 WDÿѭӧc nguyên hàm nào? dx , bҾQJFiFKÿҝt u x 1 A.

Âł

A. Âł 2u u 2  4 du .

&kX 10: HĂ m sÓ&#x2022; F x

f x

B.

Âł u

2

ax  b

 4 du .

C. Âł 2 u 2  4 du .

D.

Âł u

2

 3 du .

4 x  1 ( a, b lĂ cĂĄc hŇľng sÓ&#x2022; thÓľc) lĂ  mÓ?t nguyĂŞn hĂ m cÓŤa

12 x . TĂ­nh a  b . 4x  1

A. 0 .

B. 1 .

C. 2 .

D. 3 .

&kX TÓ&#x2022;FĂżÓ? WKD\ĂżÓ&#x2122;i doanh thu (bŇľQJĂż{ODWUrQPÓ?t mĂĄy tĂ­nh) cho viÓ&#x2039;c bĂĄn x mĂĄy tĂ­nh lĂ f x , biÓ&#x192;t f c x 12 x 5  3 x 2  2 x  12 . TĂŹm tÓ&#x2122;ng doanh WKXNKLEiQÿѭӧFPŃ­ÓĄLKDLPi\WtQKÿҍu tiĂŞn. A. Ăż{OD

B. Ăż{OD

C. Ăż{OD

272

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D. Ăż{OD


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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ?

&kX MÓ?t bĂĄc thÓ§ [k\EŃŤPQŃ­Ó&#x;c vĂ o bÓ&#x2021; chÓ­DQŃ­Ó&#x;c. GÓ&#x2018;i h t lĂ  thÓ&#x2021; WtFKQŃ­Ó&#x;FEŃŤPÿѭӧc sau t giây. Cho hc t 3at 2  bt YjEDQÿҍu bÓ&#x2021; NK{QJFyQŃ­Ó&#x;c. Sau 5 giây thĂŹ thÓ&#x2021; WtFKQŃ­Ó&#x;c trong bÓ&#x2021; lĂ  150m3 , sau 10 giây thĂŹ thÓ&#x2021; WtFKQŃ­Ó&#x;c trong bÓ&#x2021; lĂ  1100m3. TĂ­nh thÓ&#x2021; tĂ­ch cÓŤDQŃ­Ó&#x;c trong bÓ&#x2021; VDXNKLEŃŤPÿѭӧc 20 giây. A. 8400m3 .

B. 2200m3 .

C. 600m3 .

D. 4200m3 .

Ä&#x2C6;Ă&#x2C6;3Ă&#x2C6;1&+8<Ă&#x2021;1Ä&#x2C6;Ó&#x201E; 1 1â&#x20AC;&#x201C;B

2â&#x20AC;&#x201C;A

11 â&#x20AC;&#x201C; A

12 â&#x20AC;&#x201C; A

3â&#x20AC;&#x201C;D

+Ń­Ó&#x;ng dŇŻn giҧi 2 x  13 &kX 7: Âł dx ( x  1)( x  2)

4â&#x20AC;&#x201C;B

§ 5

5â&#x20AC;&#x201C;B

6â&#x20AC;&#x201C;A

7â&#x20AC;&#x201C;D

1 1 dx  3Âł dx x 1 x2

3 ¡

5Âł

2 4x  1

6ax  a  2b . 4x  1

³ ¨Š x  1  x  2 ¸š dx

8â&#x20AC;&#x201C;B

9â&#x20AC;&#x201C;C

10 â&#x20AC;&#x201C; B

5ln x  1  3ln x  2  C .

­a 5 Vҹy Ž nên a  b 8 . ¯b 3

ChÓ&#x2018;n D. &kX 10: Ta cĂł F c x a 4 x  1  ax  b .

Ä&#x2C6;Ó&#x2021; F x lĂ mÓ?t nguyĂŞn hĂ m cÓŤa f x thĂŹ

­6a 12 ­a 2 12 x . Â&#x153;ÂŽ Â&#x153;ÂŽ 4x  1 ÂŻa  2b 0 ÂŻb 1

6ax  a  2b 4x  1

'RĂży a  b 1 . ChÓ&#x2018;n B. &kX f x

Âł 12 x

5

 3 x 2  2 x  12 dx

12 6 1 3 1 2 x  3. x  2. x  12 x  C 5 1 2 1 11

2 x 6  x3  x 2  12 x  C . 9u Ăżk\ Oj  ÂłWÓ&#x2022;F ĂżÓ? WKD\ ĂżÓ&#x2122;i doanh thu (bŇľQJ Ăż{ OD WUrQ PÓ?t mĂĄy tĂ­nh) cho viÓ&#x2039;c bĂĄn x mĂĄy tĂ­nhâ&#x20AC;? nĂŞn C 0. 'RYŇąy ta cŇŤn thay x 12 vĂ o sÓ ĂżŃ­Ó§c f 12 2.126  123  122  12.12 5973984 . ChÓ&#x2018;n A. &kX 7DFy h t

Âł hc t dt

Âł 3at

2

 bt dt

at 3  b

'REDQÿҍu hÓ&#x2014; NK{QJFyQŃ­Ó&#x;c nĂŞn h 0 0 Â&#x153; C /~FJLk\ h 5 a.53  b. Suy ra a 1, b

52 2

t2  C. 2

0 Â&#x; h t at 3  b

150 /~FJLk\ h 10 a.103  b.

2 Â&#x; h t t 3  t 2 Â&#x; h 20 203  202

t2 . 2

102 2

1100 .

8400m3 .

ChÓ&#x2018;n A. 273

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ?

&+8<Ă&#x2021;1Ä&#x2C6;Ó&#x201E; 2

TĂ?CH PHĂ&#x201A;N A. KIÓ&#x201A;N THÓŹC TRÓ?NG TĂ&#x201A;M .KiLQLÓ&#x2039;PWtFKSKkQ

f x liĂŞn tÓŠc trĂŞn K vĂ a, b Â? K . NÓ&#x192;u F x lĂ  mÓ?t nguyĂŞn hĂ m cÓŤa y

Cho hĂ m sÓ&#x2022; y

K thĂŹ F b  F a ÿѭӧc gÓ&#x2018;i lĂ tĂ­ch phân cÓŤa y

f x tÓŻ a ĂżÓ&#x192;n b vĂ kĂ­ hiÓ&#x2039;u lĂ 

f x trĂŞn

b

Âł f x dx a

b

Âł f x dx

F b  F a .

a

ChĂş Ă˝: 7tFKSKkQ[iFĂżÓ?nh khĂ´ng phÓŠ thuÓ?c biÓ&#x192;n

b

Âł

b

f x dx

Âł f t dt

a

F b  F a .

a

1.2. TĂ­nh chŇŠt cÓŤDWtFKSKkQ a

Âł

b

Âł

f ( x)dx 0

a

a

a

Âł kf ( x)dx

b

b

b

b

a

a

a

Âł > f ( x) r g ( x)@dx Âł f ( x)dx r Âł g ( x)dx Ć&#x201D;NÓ&#x192;u f x t 0 trĂŞn > a; b @ thĂŹ

b

 Âł f ( x)dx

f ( x)dx

a

b

k Âł f ( x)dx a

b

c

b

a

a

c

Âł f ( x)dx Âł f ( x)dx  Âł f ( x)dx

b

Âł f ( x)dx t 0 . a

b

Ć&#x201D; NÓ&#x192;u f x t g x trĂŞn > a; b @ thĂŹ

Âł a

b

f ( x)dx t Âł g ( x)dx . a

B. CĂ C DҤNG BĂ&#x20AC;I TŇ°P DŇĽQJ7tQKWtFKSKkQEŇľQJSKŃ­ŃŤQJSKiSĂżÓ&#x2122;LELÓ&#x192;n sÓ&#x2022; 3KŃ­ŃŤQJSKiSJLҧL Ä&#x2C6;Ňťt t

t x . CĂ´ng thÓ­FĂżÓ&#x2122;i biÓ&#x192;n sÓ&#x2022;

b

Âł a

f ªt x ºŸ .t c x dx

t b

Âł f t dt .

t a

2. VĂ­ dÓŠ mŇŻu VĂ­ dÓŠ 1: Cho I

3

Âłx 1

A. I

2

2dt Âł t 2  1. 2

dx 1 x

2

Ä&#x2C6;Ňťt t B. I

2

1  x 2 thÏ tích phân I trӣ thà nh

tdt Âł t 2  1. 2

C. I

2

dt Âł t 2  1. 2

LÓĄi giҧi chi tiÓ&#x192;t

274

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D. I

2

2tdt . 2 1

Âłt 2


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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? CĂĄch 1: I

3

Âłx 1

Ä&#x2C6;Ňťt t

3

dx 1 x

Âłx

2

1

xdx 2

1  x2

.

1  x 2 Â&#x; t 2 1  x 2 Â&#x; 2tdt

Ä&#x2C6;Ó&#x2122;i cŇąQ x 1 Â&#x; t .KLĂży I

2 ;x

2

2

tdt Âł t 2  1 t 2

Âłt 2

2 xdx Â&#x; tdt

3Â&#x;t

xdx;

2.

dt . 1

2

CĂĄch 2: SÓą dÓŠQJPi\WtQK&DVLR 3

1

ÂłX

dX WDÿѭӧc kÓ&#x192;t quҧ lĂ 0,332067 . 1  X2 %˹ͣc 2: TĂ­nh tĂ­ch phân cÓŤDĂżiSiQĂżiSiQQjRÿѭӧc kÓ&#x192;t quҧ QKŃ­EŃ­Ó&#x;c 1 thĂŹ chÓ&#x2018;n.

%˹ͣc 1: Nhҹp tích phân

1

Ä&#x2C6;iSiQ$QKŇąp I

2

ÂłX 2

Ä&#x2C6;iSiQ%QKŇąp I

2

XdX WDÿѭӧc kÓ&#x192;t quҧ lĂ 0,549306 , loŇĽLĂżiSiQ% 2 1

ÂłX 2

Ä&#x2C6;iSiQ&QKŇąp I

2 dX WDÿѭӧc kÓ&#x192;t quҧ lĂ 0,664134 , loŇĽLĂżiSiQ$ 1

2

2

ÂłX

1 dX WDÿѭӧc kÓ&#x192;t quҧ lĂ 0,332067 , chÓ&#x2018;QĂżiSiQ& 1

2

2

ChÓ&#x2018;n C. VĂ­ dÓŠ 2: Cho I

1

x3 Âł0 x 4  1 dx

3 . 4 LÓĄi giҧi chi tiÓ&#x192;t A. a  b

CĂĄch 1: Ä&#x2C6;Ňťt u

a  b.ln 2 WURQJĂżya, b lĂ cĂĄc sÓ&#x2022; nguyĂŞn. ChÓ&#x2018;n phĂĄt biÓ&#x2021;XĂż~QJ.

B. a  b

4

x  1 Â&#x; du

3

1 . 4

4 x dx . I

C. a  b 2

du

Âł 4u 1

1 nĂŞn a  b 4 CĂĄch 2: SÓą dÓŠQJYLSKkQ 'RĂży a 0; b

I

1

x3 Âł0 x 4  1 dx

1

d x 4  1

Âł 4 x 0

4

D. a b.

1 1 1 ln 2  ln1 ln 2. 4 4 4

1 . 4 1 1 .ln x 4  1 0 4

 1

1 2 ln | u | 1 4

1 . 2

1 1 1 ln 2  ln1 ln 2 . 4 4 4

1 1 nĂŞn a  b . 4 4 CĂĄch 3: SÓą dÓŠQJPi\WtQK&DVLR 'RĂży a 0; b

%˹ͣc 1: Nhҹp tích phân I

1

Âł 0

X3 dX ÿѭӧc kÓ&#x192;t quҧ lĂ 0,1732867951 . X 4 1

%˹ͣc 2: GĂĄn giĂĄ trÓ? kÓ&#x192;t quҧ FKR$6+,)7672$ 275

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? %˹ͣc 3: Giҧi hÓ&#x2039; SKŃ­ŃŤQJWUuQKKDLŇ­n ­a  b ln 2 ° Ä&#x2C6;iSiQ$1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUuQK ÂŽ 3 °Ě&#x201E;a  b 4 ­a  b ln 2 ° Ä&#x2C6;iSiQ%1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUuQK ÂŽ 1 °Ě&#x201E;a  b 4

­a  b ln 2 ° Ä&#x2C6;iSiQ&1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUuQK ÂŽ 1 °Ě&#x201E;a  b 2

A

A

A

ÿѭӧc kÓ&#x192;t quҧ lĂ nghiÓ&#x2039;m vĂ´ tÓ?, loŇĽi.

­a 0 ° ÿѭӧc kÓ&#x192;t quҧ lĂ ÂŽ 1 , chÓ&#x2018;QĂżiSiQ% °Ě&#x201E;b 4 ÿѭӧc kÓ&#x192;t quҧ lĂ  nghiÓ&#x2039;m vĂ´ tÓ?, loŇĽi.

­a  b ln 2 A ÿѭӧc kÓ&#x192;t quҧ lĂ nghiÓ&#x2039;m vĂ´ tÓ?, loŇĽi. Ä&#x2C6;iSiQ'1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUunh ÂŽ ÂŻa  b 0 ChÓ&#x2018;n B. 3 x a VĂ­ dÓŠ 3: Cho Âł dx  b ln 2  c ln 3 vÓ&#x;i a , b , c lĂ  cĂĄc sÓ&#x2022; nguyĂŞn. 3 0 4  2 x 1

GiĂĄ trÓ? cÓŤa a  b  c bŇľng A. 1 . B. 2 . LÓĄi giҧi chi tiÓ&#x192;t Ä&#x2C6;Ňťt t x  1 Â&#x; t 2 x  1 Â&#x; x t 2  1 Â&#x; dx Ä&#x2C6;Ó&#x2122;i cŇąQ x 0 Â&#x; t 1 ; x 3 Â&#x; t 2 . .KLĂży 2

t2 1 Âł1 4  2t .2tdt

2

2

t3  t Âł1 t  2 dt

6 ¡ § 2 ³1 ¨Š t  2t  3  t  2 ¸š dt

C. 7 .

D. 9 .

2tdt .

2

§ t3 2 ¡ ¨  t  3t  6ln t  2 ¸ Š3 š1

­a 7 ° Suy ra ÂŽb 12 Â&#x; a  b  c 1 . °c 6 ÂŻ ChÓ&#x2018;n A.

DҼQJ7tFKSKkQWӯQJSKҍn 3KѭэQJSKiSJLҧL b

b

b

NÓ&#x192;u u vĂ v lĂ  hai hĂ m sÓ&#x2022; FyÿҼo hĂ m liĂŞn tÓŠc trĂŞn K Âł udv uv a  Âł vdu. a

2. Ví dӊ mүu Ví dӊ 1: Cho tích phân I 2

2

2

ln x dx . KhҡQJĂżÓ?QKQjRVDXĂżk\Ăż~QJ" x2 1

Âł

A. I

1 1  ln x  Âł 2 dx. x x 1 1

C. I

1 2  ln x 1  Âł dx. x 1

2

a

2

2

B. I

1 1  ln x  Âł 2 dx. x x 1 1

D. I

1 2  ln x 1  Âł dx. x 1

2

276

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? LÓĄi giҧi chi tiÓ&#x192;t 1 ­ du dx ­u ln x ° ° ° x 'RĂży I Ä&#x2C6;Ňťt ÂŽ Â&#x; 1 ÂŽ °¯dv x 2 dx °v 1 °Ě&#x201E; x

2

2

1 1  ln x  Âł 2 dx. x x 1 1

ChÓ&#x2018;n B. 2

Âł (4 x  3).ln xdx

Ví dӊ 2: Tính tích phân I

a ln 2  b . TĂ­nh giĂĄ trÓ? cÓŤa a  2b .

1

A. 1.

B. 1 .

C. 2.

D.

1 . 2

LÓĄi giҧi chi tiÓ&#x192;t  ­ dx °­u ln x °du CĂĄch 1: Ä&#x2C6;Ňťt ÂŽ . Â&#x;ÂŽ x °Ě&#x201E;dv 4 x  3 dx °v 2 x 2  3 x ÂŻ

.KLĂży I

2

2 x 2  3x dx x 1

2 x2  3x ln x  Âł 2

1

14ln 2  0  x 2  3 x

2 1

2

2.22  3.2 ln 2  2.12  3.1 ln1  Âł 2 x  3 dx 1

14ln 2  0  ª 22  3.2  12  3.1 ºŸ

14ln 2  10  4 14ln 2  6. 'RĂży a 14; b 6 Â&#x; a  2b 14  2 6 2 . CĂĄch 2: SÓą dÓŠQJPi\WtQK&DVLR %˹ͣc 1: NhŇąp tĂ­ch phân

2

Âł 4X  3 .ln X dX ÿѭӧc kÓ&#x192;t quҧ lĂ 3,704060528. 1

%˹ͣc 2: GĂĄn giĂĄ trÓ? kÓ&#x192;t quҧ FKR$6+,)7672$ %˹ͣc 3: Giҧi hÓ&#x2039; SKŃ­ŃŤQJWUuQKKDLŇ­n ­a ln 2  b A Ä&#x2C6;iSiQ$1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUuQK ÂŽ ÿѭӧc kÓ&#x192;t quҧ lĂ nghiÓ&#x2039;m vĂ´ tÓ?, loŇĽi. ÂŻa  2b 1 ­a ln 2  b A ÿѭӧc kÓ&#x192;t quҧ lĂ  nghiÓ&#x2039;m vĂ´ tÓ?, loŇĽi. Ä&#x2C6;iSiQ%1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUuQK ÂŽ ÂŻa  2b 1 ­a ln 2  b A ­a 14 Ä&#x2C6;iSiQ&1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUuQK ÂŽ ÿѭӧc kÓ&#x192;t quҧ lĂ  ÂŽ , chÓ&#x2018;QĂżiSiQ& ÂŻa  2b 2 ÂŻb 6

­a ln 2  b A ° ÿѭӧc kÓ&#x192;t quҧ lĂ nghiÓ&#x2039;m vĂ´ tÓ?, loŇĽi. Ä&#x2C6;iSiQ'1KŇąp hÓ&#x2039; SKŃ­ŃŤQJWUuQK ÂŽ 1 °Ě&#x201E;a  2b 2 ChÓ&#x2018;n C.

277

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ?

DŇĽng 37tFKSKkQPÓ?t sÓ&#x2022; KjPWKŃ­ÓĄng gŇťS S 3

Âł sin

VĂ­ dÓŠ 1: Cho I

b x tan xdx ln a  . ChÓ&#x2018;n mÓ&#x2039;QKĂżÓ&#x2026; Ăż~QJ 8

2

0

B. a  b

A. a  b 4. LÓĄi giҧi chi tiÓ&#x192;t

I

S 3

Âł sin

2

Ä&#x2C6;Ňťt u

I

1 2

Âł

1  u du

2

1

1

§ u2 ¡ §1 ¡ ln   u du u ¸ ³1 ¨Š u ¸š ¨Š 2 š1

u

cos x

0

1 , x 0 Â&#x; u 1. 2 1¡ § 1 1¡ 3 § ¨ ln1  ¸  ¨ ln  ¸ ln 2  . 2š Š 2 8š 8 Š

2

2

4.

2

Âł

S Â&#x;u 3

D. a b

1  cos x sin x dx .

3

sin x Âł0 cos x dx

2

sin xdx Ä&#x2C6;Ó&#x2122;i cŇąn x

cos x Â&#x;  du

1

3

3

sin x Âł0 sin x. cos x dx

x tan xdx

S

S

S 3

0

C. ab 6.

2.

2, b 3 nĂŞn ab 6.

'RĂży a ChÓ&#x2018;n C.

4

Âłx

VĂ­ dÓŠ 2: %LÓ&#x192;t I

3

dx x

a ln 2  b ln 3  c ln 5, vÓ&#x;i a, b, c lĂ cĂĄc sÓ&#x2022; nguyĂŞn. TĂ­nh S

2

B. S

A. S 6 . LÓĄi giҧi chi tiÓ&#x192;t 4

dx Âł3 x 2  x

CĂĄch 1: I

4

dx Âł3 x x  1

C. S

2. 4

x 1 x Âł3 x x  1 dx

ln 4  ln 5  ln 3  ln 4

4, b 1, c 1 Â&#x; S

6X\UD a

I

1

Âł x  a x  b dx

4, b 1, c 1 , tÓŻ Ăży S

6X\UD a

4

ªln x  ln x  1 Ÿº 3

2.

abc

1 ln x  ln x  1

0 1 3

dx Âł3 x x  1

1 ¡

3

4

4

dx Âł3 x 2  x

§1

³ ¨Š x  x  1 ¸š dx

0.

4ln 2  ln 3  ln 5.

CĂĄch 2: Ă p dÓŠng cĂ´ng thÓ­c tĂ­nh nhanh 4

4

D. S

2 .

a  b  c.

abc

1 ln x  a  ln x  b WDÿѭӧF a b

ln 4  ln 5  ln 3  ln 4

4ln 2  ln 3  ln 5 .

2.

CĂĄch 3: SÓą dÓŠQJPi\WtQK&DVLR 4

Âłx

%˹ͣc 1: I

3

a

b

'RĂży 2 .3 .5c

dx x

2

a ln 2  b ln 3  c ln 5 ln 2a  ln 3b  ln 5c

ln 2a.3b.5c .

eI .

%˹ͣc 2: Tính tích phân

4

Âłx 3

2

1 dx bŇľng mĂĄy tĂ­nh, gĂĄn giĂĄ trÓ? WKXÿѭӧFFKR$6KLIW672$ x

%˹ͣc 3: TĂ­nh e A WDÿѭӧc kÓ&#x192;t quҧ lĂ

16 15

24.31.51 GRĂży a 4, b 1, c 1 .

ChÓ&#x2018;n B. 278

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? a

2 ? 7

VĂ­ dÓŠ 3: &yEDRQKLrXVÓ&#x2022; a Â? 0;20S sao cho Âł sin 5 x.sin 2 xdx 0

A. 20. LÓĄi giҧi chi tiÓ&#x192;t

B. 19.

a

Ta cĂł Âł sin 5 x sin 2 xdx 0

C. 9.

a

2 Â&#x153; 2 Âł sin 6 x cos xdx 7 0 a

sin 7 x Â&#x153; 7 0

VĂŹ a Â? 0;20S Â&#x; 0 

D. 10.

a

2 Â&#x153; Âł sin 6 d sin x

7 0

1 7

1 Â&#x153; sin 7 a 1 Â&#x153; sin a 1 Â&#x153; a 7

S 2

 k 2S k Â? ] .

S S 39S 1 39  k 2S  20S Â&#x153;   k 2S  Â&#x153; k , 2 2 2 4 4

k Â? ] Â&#x; k Â? ^0;1;2;3;4;5;6;7;8;9`. VŇą\FyVÓ&#x2022;a WKÓ&#x201C;DPmQĂżÓ&#x2026;EjL ChÓ&#x2018;n D.

DҼng 47tFKSKkQKjPҭn 3KѭэQJSKiSJLҧL b

b

7tFKSKkQ[iFĂżÓ?nh khĂ´ng phÓŠ thuÓ?c biÓ&#x192;n Âł f x dx

f t dt

Âł

a

a

b

Âł f u du . a

2. VĂ­ dÓŠ mŇŻu VĂ­ dÓŠ 1: Cho

4

Âł

f x dx 18,

2

8

Âł

f x dx 15 %LÓ&#x2021;u thÓ­c

2

A. 3.

8

Âł f x dx

bŇľng

4

B. 33.

C. 3 .

D. 33 .

LÓĄi giҧi chi tiÓ&#x192;t Ta cĂł

8

Âł

f x dx

2

4

Âł 2

8

8

4

4

f x dx  Âł f x dx Â&#x; Âł f x dx

8

4

f x dx  Âł f x dx 15  18 3 .

Âł 2

2

ChÓ&#x2018;n C. VĂ­ dÓŠ 2: Cho

2

Âł x  2 f c x dx

5 vĂ f 0 1. TĂ­nh I

0

A. I

3.

B. I

2

Âł f x dx . 0

C. I

3.

D. I

7.

7.

LÓĄi giҧi chi tiÓ&#x192;t ­u x  2 ­du dx .KLĂży Ä&#x2C6;Ňťt ÂŽ Â&#x;ÂŽ ÂŻdv f c( x)dx ÂŻv f ( x)

5

2

Âł x  2 f c x dx 0

2

( x  2) f ( x) 0  Âł f x dx 2

2 f (0)  I

2I Â&#x; I

3.

0

ChÓ&#x2018;n B.

279

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? VĂ­ dÓŠ 3: Cho

2

Âł

f x 2  1 xdx

5

Âł f x dx bŇľng

2 .KLĂży I

1

2

A. 2 . LÓĄi giҧi chi tiÓ&#x192;t

B. 1 .

C. 1 .

x 2  1 Â&#x; dt 2 xdx . Ä&#x2C6;Ó&#x2122;i cŇąQ x 1 Â&#x; t 2 , x 2 Â&#x; t

D. 4 .

Ä&#x2C6;Ňťt t

2

5.

5

5

2

1 f t dt Â&#x; Âł f t dt 2 Âł2 2

.KLĂży Âł f x 2  1 xdx 1

MĂ tĂ­ch phân khĂ´ng phÓŠ thuÓ?c vĂ o biÓ&#x192;QQrQ I

2 Âł f x 2  1 xdx

4.

1

5

Âł

5

f x dx

Âł f t dt

2

4.

2

ChÓ&#x2018;n D. VĂ­ dÓŠ 4: Cho hĂ m sÓ&#x2022; f c x thÓ&#x201C;a mĂŁn f 2 

f 1 bŇľng 41 . 100 LÓĄi giҧi chi tiÓ&#x192;t

B.

A.

1 . 10

ª f x ºŸ

ÂŹ

2 vĂ o hai vÓ&#x192; WDFy

1 f x

Âź

2

dx

D.

1

Âł 4 x dx Â&#x153;  f ( x)

1 16  C Â&#x153; C 1 25

x 4  9 GRĂży 

391 . 400

1 . 40

4 x3 .

2

f c x

Âł ÂŞ f x Âş

/ŇŠy nguyĂŞn hĂ m hai vÓ&#x192; ta cĂł

VŇąy 

C.

f c x

2

Ta cĂł f c x 4 x 3 . ÂŞÂŹ f x ºŸ Â&#x153;

Thay x

1 2 vĂ f c x 4 x 3 . ÂŞÂŹ f x ºŸ vÓ&#x;i mÓ&#x2018;i x Â? \ . GiĂĄ trÓ? cÓŤa 25

3

x4  C .

9.

1 1 1  9 10 Â&#x153; f 1  . f 1

10

ChÓ&#x2018;n B.

VĂ­ dÓŠ 5: Cho hĂ m sÓ&#x2022; f x Fy ÿҼR KjP GŃ­ŃŤQJ OLrQ WÓŠF WUrQ ĂżRŇĽn > 0;1@ thÓ&#x201C;a mĂŁn f 0 1 vĂ  1

1

2 1º ª 3³  f c x ª f x ºŸ   dx d 2 ³ 9Ÿ 0 0

3 . 2 LÓĄi giҧi chi tiÓ&#x192;t A.

1

B.

f c x f x dx . Tính tích phân

1

³ ª f x ºŸ

3

dx .

0

5 . 4

C.

5 . 6

D. 1

7 . 6

2 2 7DFy Âł ÂŤÂŞ 3 f c x f x  2.3 f c x f x  1º dx d 0 Â&#x153; Âł ÂŞ3 f c x f x  1Âş dx d 0 . ÂŹ Âź Âź 0ÂŹ 0

Suy ra 3 f c x f x  1 0 Â&#x153;

f c x f x

1 Â&#x153; f c x . f 2 x

3

1 . 9

280

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? VĂŹ ÂŞÂŹ f 3 x ºŸc

3. f 2 x f c x nên suy ra ª f 3 x ºŸc

1 Â&#x; f 3 x

3

1 xC. 3

VĂŹ f 0 1 nĂŞn f 3 0 1 Â&#x; C 1 .

1 x 1. 3

VŇąy f 3 x

Suy ra

1

3

³ ª f x ºŸ dx 0

1

§1

¡

³ ¨Š 3 x  1¸š dx 0

7 . 6

ChÓ&#x2018;n D. VĂ­ dÓŠ 6: Cho hĂ m sÓ&#x2022; f x thÓ&#x201C;a mĂŁn ÂŞÂŹ f c x ºŸ  f x . f cc x 15 x 4  12 x , x Â? \ 2

vĂ f 0

f c 0 1 . GiĂĄ trÓ? cÓŤa f 2 1 bŇľng

9 . 2 LÓĄi giҧi chi tiÓ&#x192;t

B.

A.

5 . 2

C. 10 .

D. 8 .

7DFy f c x  f x . f cc x 15 x 4  12 x , x Â? \ . 2

Â&#x153; ÂŞÂŹ f c x . f x ټc 15 x 4  12 x , x Â? \ Â&#x153; f c x . f x 3 x5  6 x 2  C1 . 'R f 0

f c 0 1 nĂŞn ta cĂł C1 1.

§1 ¡c TÓŻ Ăży f c x . f x 3 x 5  6 x 2  1 Â&#x153; ¨ f 2 x ¸ Š2 š MĂ f 0 1 nĂŞn ta cĂł C2 1. 'RĂży f 2 x

3x5  6 x 2  1 Â&#x153; f 2 x

x 6  4 x 3  2 x  C2 .

x6  4 x3  2 x  1 .

VŇąy f 2 1 8.

ChÓ&#x2018;n D. DŇĽng 5: ÓŹng dÓŠQJWtFKSKkQWURQJQKÓłQJEjLWRiQWKÓľc tÓ&#x192; 3KŃ­ŃŤQJSKiSJLҧL GÓ&#x2018;i v t , s t , a t lŇŤQOŃ­Ó§t lĂ vŇąn tÓ&#x2022;FTXmQJÿѭӥng, gia tÓ&#x2022;c cÓŤa vŇąt, ta cĂł mÓ&#x203A;i liĂŞn hÓ&#x2039; giÓłDFK~QJ QKŃ­VDX Âł v t dt

s t

Âł a t dt

v t

b

4XmQJÿѭӥng vҹWÿLÿѭӧc trong khoҧng thӥi gian t � > a; b @ là ³ v t dt

s b  s a .

a

2. VĂ­ dÓŠ mŇŻu VĂ­ dÓŠ 1: MÓ?t chŇŠW ĂżLÓ&#x2021;P ĂżDQJ FKX\Ó&#x2021;Q ĂżÓ?ng vÓ&#x;i vŇąn tÓ&#x2022;c v0 15 m/s WKu WÄ&#x192;QJ WÓ&#x2022;c vÓ&#x;i gia tÓ&#x2022;c

a t t 2  4t m/s 2 . 7tQKTXmQJÿѭӥng chŇŠWĂżLÓ&#x2021;PĂżyĂżLÿѭӧc trong khoҧng thÓĄi gian 3 giây kÓ&#x2021; tÓŻ O~F

bŇłWÿҍXWÄ&#x192;QJ tÓ&#x2022;c. A. 70, 25 m .

B. 68, 25 m .

C. 67, 25 m .

D. 69,75 m . 281

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? LÓĄi giҧi chi tiÓ&#x192;t

a t t 2  4t Â&#x; v t

MĂ v 0 C 15 Â&#x; v t

C Â? \ .

t3  2t 2  15 . 3

3

§ t3 ¡ 2 ³0 ¨Š 3  2t  15 ¸š dt

VŇąy S

t3  2t 2  C 3

Âł a t dt

69,75 m .

ChÓ&#x2018;n D. VĂ­ dÓŠ 2: MÓ?t Ă´ tĂ´ chuyÓ&#x2021;QĂżÓ?ng nhanh dŇŤQĂżÓ&#x2026;u vÓ&#x;i vŇąn tÓ&#x2022;c v t 7t m/s Ä&#x2C6;Lÿѭӧc 5 s QJŃ­ÓĄi lĂĄi xe phĂĄt hiÓ&#x2039;Q FKŃ­Ó&#x;ng ngŇĽi vŇąt vĂ phanh gŇŠp, Ă´ tĂ´ tiÓ&#x192;p tÓŠc chuyÓ&#x2021;Q ĂżÓ?ng chŇąm dŇŤQ ĂżÓ&#x2026;u vÓ&#x;i gia tÓ&#x2022;c a

35 m / s 2 . 7tQKTXmQJÿѭӥng cÓŤD{W{ĂżLÿѭӧc tÓŻ O~FEŇłWÿҍu chuyÓ&#x2021;QEiQKFKRĂżÓ&#x192;n khi dÓŻng hҡn.

B. 96.5 m.

A. 87.5 m. LÓĄi giҧi chi tiÓ&#x192;t

C. 102.5 m. 5

4XmQJÿѭӥQJ{W{ÿLÿѭӧc trong 5 s ÿҍu là s1

Âł 7tdt 0

t2 7 2

D. 105 m.

5

87,5 m. 0

3KŃ­ŃŤQJWUuQKYŇąn tÓ&#x2022;c cÓŤa {W{NKLQJŃ­ÓĄi lĂĄi xe phĂĄt hiÓ&#x2039;QFKŃ­Ó&#x;ng ngŇĽi vŇąt lĂ  v 2 t 35  35t (m/s). Khi xe dÓŻng lŇĽi hҡn thĂŹ v 2 t 0 Â&#x153; 35  35t

0 Â&#x153; t 1.

4XmQJÿѭӥQJ{W{ĂżLÿѭӧc tÓŻ khi phanh gŇŠSĂżÓ&#x192;n khi dÓŻng lŇĽi hҡn lĂ 1

1

Âł 35  35t dt

s2

0

§ t2 ¡ 35 t  35 ¨ ¸ 17.5 (m). 2 š0 Š

VŇą\TXmQJÿѭӥng cÓŤD{W{ĂżLÿѭӧc tÓŻ O~FEŇłWÿҍu chuyÓ&#x2021;QEiQKFKRĂżÓ&#x192;n khi dÓŻng hҡn lĂ

s

s1  s2

87.5  17.5

105 (m).

ChÓ&#x2018;n D.

BĂ&#x20AC;I TŇ°P TÓ&#x2DC;NG HÓŚ3&+8<Ă&#x2021;1Ä&#x2C6;Ó&#x201E; 2 &kX Cho I

2

Âł

f x dx 3 .KLĂży J

0

A. 2 .

2

³ ª 4 f x  3ºŸ dx bҾng 0

B. 6 .

&kX .KLĂżÓ&#x2122;i biÓ&#x192;n x

C. 8 .

3 tan t , tích phân I

1

Âłx 0

A. I

S 3

Âł

3dt .

0

B. I

S 6

Âł 0

D. 4 .

dx trӣ thà nh tích phân nào? 3

2

S

S

3 dt 3

C. I

6

Âł

3tdt .

0

D. I

6

1

Âł t dt . 0

&kX  %ŇĽn Minh ngÓ&#x2014;L WUrQ Pi\ ED\ ĂżL GX OÓ?ch thÓ&#x192; giÓ&#x;i vĂ vŇąn tÓ&#x2022;c chuyÓ&#x2021;Q ĂżÓ?ng cÓŤa mĂĄy bay lĂ  v t 3t 2  5 (m/s) 7tQKTXmQJÿѭӥQJPi\ED\ĂżLÿѭӧc tÓŻ giây thÓ­ 4 ĂżÓ&#x192;n giây thÓ­ 10 . A. 246 m .

B. 252 m .

C. 1134 m .

282

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D. 966 m .


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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? 2

b b  a ln 2 (vÓ&#x;i a lĂ sÓ&#x2022; thÓľc, b , c lĂ  cĂĄc sÓ&#x2022; QJX\rQGŃ­ŃŤQJYj lĂ  phân sÓ&#x2022; tÓ&#x2022;i c c

ln x dx x2 1

Âł

&kX %LÓ&#x192;t

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D. 11 . 283

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GOODA.VN - THU VIáť&#x2020;N SĂ CH QUĂ? 2

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G O O D A . V N - T Háť&#x2020;N U VSĂ C I H QU Ă?

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285

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GOODA.VN - THU VIỆN SÁCH QUÝ

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GOODA.VN - THU VIỆN SÁCH QUÝ

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B. M c 1;  1 .

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Lӡi giҧi chi tiӃt

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360

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D. 'c  x   y    .


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GOODA.VN - THU VIỆN SÁCH QUÝ Lӡi giҧi chi tiӃt Gӑi M x; y OjÿLӇPWKXӝc ' .

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.

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DҥQJ;iFÿӏnh phép tӏnh tiӃn khi biӃt ҧnh và tҥo ҧnh 3KѭѫQJSKiSJLҧi

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Lӡi giҧi chi tiӃt

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GOODA.VN - THU VIỆN SÁCH QUÝ G Câu 5: &KR v

1; 5

YjÿLӇm M c 4; 2 . BiӃW M c là ҧnh cӫD M TXDSKpSWӏQKWLӃn TvG . Tìm M .

A. M  .

B. M 3;5 .

G Câu 6: 7URQJPһWSKҷQJWӑDÿӝ Oxy FKR v G SKpSWӏQKWLӃQWKHRYHFWѫ v .

C. M 3; 7 .

D. M 5;3 .

2; 3 +m\WuPҧnh cӫDFiFÿLӇm A 1; 1 , B 4; 3

A. A' 1; 2 , B 2; 6 .

B. A' 1; 2 , B 2; 6 .

C. A' 1; 2 , B 2; 6 .

D. A' 1;1 , B 2; 6 .

TXD

Câu 7: &KRSKpSELӃn hình F FyTX\WҳFÿһWҧQKWѭѫQJӭQJÿLӇm M xM ; yM FyҧQKOjÿLӇm M ' x'; y'

­ x' WKHRF{QJWKӭc F : ® ¯ y'

xM yM  1

7tQKÿӝ GjLÿRҥQWKҷng PQ vӟi P, Q WѭѫQJӭng là ҧnh cӫDKDLÿLӇm

A  ,B  TXDSKpSELӃn hình F. A. PQ

B. PQ

2.

C. PQ 3 2 .

2 2.

D. PQ

4 2.

Câu 8: &KRWDPJLiF ABC và I , J lҫQOѭӧWOjWUXQJÿLӇm cӫD AB, AC 3KpSELӃn hình T ELӃQÿLӇm JJJJJG JJG M WKjQKÿLӇm M ' VDRFKR MM ' 2 IJ . MӋQKÿӅ QjRVDXÿk\ÿ~QJ" JJG JJG B. T OjSKpSWӏQKWLӃQWKHRYHFWѫ  IJ . A. T OjSKpSWӏQKWLӃQWKHRYHFWѫ IJ . JJJG JJJG C. T OjSKpSWӏQKWLӃQWKHRYHFWѫ CB . D. T OjSKpSWӏQKWLӃQWKHRYHFWѫ BC . JG Câu 9: 7URQJPһWSKҷng Oxy FKRYHFWѫ v 3;2 YjÿѭӡQJWKҷng ' : x   y    . ViӃWSKѭѫQJ JG WUuQKÿѭӡQJWKҷng 'c là ҧnh cӫDÿѭӡQJWKҷng ' TXDSKpSWӏQKWLӃQWKHRYHFWѫ v . A. 'c :  x  y    .

B. 'c :  x  y    .

C. 'c : x   y    .

D. 'c  [   \   

Câu 10: 7URQJ PһW SKҷQJ WӑD ÿӝ Oxy  YLӃW SKѭѫQJ WUuQK ÿѭӡQJ WUzQ C ' là ҧnh cӫD ÿѭӡQJ WUzQ G C  x 2  y 2   x   y    TXDSKpSWӏQKWLӃQWKHRYHFWR v 1;3 . A. &  [    \  

2

2

C. &  [    \  

2

2

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2–B

3–D

4–A

5–D

6–A

7–B

8–D

9–D

10 – B

Câu 4: ĈѭӡQJWUzQ C1 FyWkP I1 1;2 ĈѭӡQJWUzQ C2 FyWkP I 2 3; 4 . JJJG G G Vì TuG ª¬ C1 ¼º C2 Ÿ TuG I1 I 2 œ I1 I 2 u Ÿ u 4; 6 . Chӑn A.

JJJG Câu 7: 7KHRTX\WҳFWDFy P 1;1 , Q 1; 3 Ÿ PQ

2; 2 Ÿ PQ

2 2.

Chӑn B.

363

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GOODA.VN - THU VIแป†N SรCH QUร Cรขu 8:

A

JJG JJJJJG JJG ฤˆาทQJWKำญc MM ' 2 IJ chำญQJWำ“ T OjSKpSWำQKWLำƒQWKHRYHFWัซ 2 ,-  7KHRJLาง WKLำƒWWDFy IJ OjรฟัญำกQJWUXQJEuQKFำซDWDPJLiF ABC QrQ JJG JJJG VX\UD 2IJ BC .

Chำ‘n D.

J

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Cรขu 9: 7DFy 'c//' ยŸ 'c : x   y  m  m z 6 . Lาฉ\ M ;  ย ' JLาง sำฑ M c TvG M ยŸ M c   ;    ยŸ M c 3; 4 . 'R M c ย 'c ยŸ     m  ยŸ m 15 WKำ“DPmQ m z 6 ยŸ 'c : x   y    . Chำ‘n D. Cรขu 10: ฤˆัญำกQJWUzQ C FyWkP I 2;1 EiQNtQK R 7DFy TvG I

22  12  1 2.

I ' 3;4 OjWkPFำซDรฟัญำกQJWUzQ C ' .

ฤˆัญำกQJWUzQ C ' FyWkP I ' 3;4 YjEiQNtQK R ' R

2 FySKัญัซQJWUuQK x  3  y  4

Chำ‘n B.

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2

2

4.


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GOODA.VN - THU VIỆN SÁCH QUÝ

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GOODA.VN - THU VIỆN SÁCH QUÝ

B. CÁC DҤNG BÀI TҰP Dҥng 1: Tìm ҧnh cӫDKuQKTXDSKpSÿӕi xӭng trөc. 3KѭѫQJSKiSJLҧi Dùng ÿӏQKQJKƭDKRһFELӇXWKӭFWӑDÿӝ cӫDSKpSÿӕi xӭQJWUөc. 2. Ví dө mүu Ví dө 1: &KRWDPJLiF ABC ÿӅX+ӓLKuQKWDPJLiF ABC FyEDRQKLrXWUөFÿӕi xӭng A. .K{QJFyWUөFÿӕi xӭng. B. &yWUөFÿӕi xӭng. C. &yWUөFÿӕi xӭng. D. &yWUөFÿӕi xӭng. Lӡi giҧi chi tiӃt WUөFÿӕi xӭng cӫDWDPJLiFÿӅXOjÿѭӡQJWUXQJWUӵc cӫDFҥnh.

Chӑn D. Ví dө 2: 7URQJPһWSKҷng Oxy FKRÿLӇm M 2; 3 . HӓLWURQJEӕQÿLӇPVDXÿLӇPQjROjҧnh cӫD M TXDSKpSÿӕi xӭQJWUөc Ox ? A. 3; 2 .

B. 2;3 .

C. 3;2 .

D. 2; 3 .

Lӡi giҧi chi tiӃt

­ xc x ­ xc 2 Gӑi M c xc; y c là ҧnh cӫDÿLӇm M x; y TXDSKpSÿӕi xӭQJWUөc Ox WDFy ® Ÿ®  ¯ y c  y ¯ y c 3 Vұ\ M c 2; 3 . Chӑn B. Ví dө 3: ;iFÿӏnh ҧnh cӫDÿѭӡQJWUzQ C : x  1  y  2

2

A. x  1  y  2

2

2

C. x  1  y  2

2

2

1 B. x  1  y  2

2

2

4 D. x  1  y  2

2

2

2

1 TXDSKpSÿӕi xӭQJWUөc Oy .

1

1

Lӡi giҧi chi tiӃt 7DFy I 1;2 OjWkPÿѭӡQJWUzQ C . ­x Gӑi I ' là ҧnh cӫD I TXDSKpSÿӕi xӭQJWUөc Oy QrQWDFy ® I ' ¯ yI '

1

 xI yI

2

I ' OjWkPÿѭӡQJWUzQ C' là ҧnh cӫD C TXDSKpSÿӕi xӭQJWUөc Oy .

366

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Ÿ I ' 1; 2 .


G O O D A . V N - T H UỆN V I SÁC H Q UÝ

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'RSKpSÿӕi xӭQJWUөFELӃQÿѭӡQJWUzQWKjQKÿѭӡQJWUzQFyFQJEiQNtQKQrQWDFySKѭѫQJWUuQK ÿѭӡQJWUzQ C'  x  1  y  2

2

2

1.

Chӑn D. Ví dө 4: 7URQJPһWSKҷng Oxy FKRÿLӇm M 2; 3 . HӓLWURQJEӕQÿLӇPVDXÿLӇPQjROjҧnh cӫD M

?

TXDSKpSÿӕi xӭQJTXDÿѭӡQJWKҷng ' : x – y A. 3; 2 .

B. 2;3 .

C. 3;2 .

D. 2; 3 .

Lӡi giҧi chi tiӃt Gӑi M c xc; y c là ҧnh cӫDÿLӇm M x; y TXDSKpSÿӕi xӭQJTXD ' : x – y Gӑi d OjÿѭӡQJWKҷQJÿLTXDÿLӇm M 2; 3 YjYX{QJJyF ' : x – y

d : x  y    . Gӑi I

.

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§5 5· d ˆ ' WKu I ¨ ; ¸ . ©2 2¹

.KLÿy I là WUXQJÿLӇm cӫD MM c QrQVX\UD M c 3; 2 . Chӑn A. Ví dө 5: &KRÿѭӡQJWKҷng d FySKѭѫQJWUuQK  x   y    YjÿѭӡQJWKҷng ' FySKѭѫQJWUuQK

x   y    3KѭѫQJWUuQKÿѭӡQJWKҷng d c là ҧnh cӫDÿѭӡQJWKҷng d TXDSKpSÿӕi xӭQJWUөc

'

A. x    .

B.  x  y    .

C.  [   \   

D. \   

Lӡi giҧi chi tiӃt

d ˆ ' Ÿ M 1; 3 . Lҩ\ N 2; 1  d . Gӑi d1 OjÿѭӡQJWKҷQJTXD N YjYX{QJJyFYӟi ' WDFy d1 :  x  y    . Gӑi I d1 ˆ ' Ÿ I 3; 1 . Gӑi N c là ҧnh cӫD N TXDSKpSÿӕi xӭQJWUөc ' QrQ I OjWUXQJÿLӇm cӫD NN c QrQ N c 4; 3 . d c là ҧnh cӫDÿѭӡQJWKҷng d TXDSKpSÿӕi xӭQJWUөc ' . 'Rÿy d c OjÿѭӡQJWKҷQJTXD M 1; 3 và N c 4; 3 . Vұ\ d c : y    . Gӑi M

Chӑn D.

Dҥng 2: Tìm ҧnh cӫDKuQKTXDSKpSÿӕi xӭng tâm. 3KѭѫQJSKiSJLҧi 'QJÿӏQKQJKƭDKRһFELӇXWKӭFWӑDÿӝ cӫDSKpSÿӕi xӭQJWkP 2. Ví dө mүu Ví dө 1: Hình gӗPKDLÿѭӡQJWUzQSKkQELӋWFyFQJEiQNtQKFyEDRQKLrXWkPÿӕi xӭng? A. .K{QJFy B. MӝW C. +DL D. Vô sӕ. 367

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GOODA.VN - THU VIỆN SÁCH QUÝ

Lӡi giҧi chi tiӃt &yPӝWWkPÿӕi xӭQJOjWUXQJÿLӇm I cӫDÿRҥQWKҷng nӕLKDLWkP

Chӑn B. Ví dө 2: 7URQJPһWSKҷng Oxy ҧnh cӫDÿLӇm A 5;3 TXDSKpSÿӕi xӭQJWkP I 4;1 là A. Ac 5;3 .

B. Ac –5; – 3 .

§ · D. Ac ¨ ;2 ¸ . ©2 ¹

C. Ac 3; – 1 .

Lӡi giҧi chi tiӃt

­ [c 24  5 3 7KD\ELӇXWKӭFWӑDÿӝ cӫDSKpSÿӕi xӭQJWkP I 4;1 WDÿѭӧF ® . ¯ \ c 21  3 1 Vұ\ A ' 3; 1 . Chӑn C. Ví dө 3: &KRÿLӇm I 1;1 YjÿѭӡQJWKҷng d : x   y    . Tìm ҧnh cӫD d TXDSKpSÿӕi xӭQJWkP ,  A. d ' : x  y    .

B. d ' : x   y    .

C. d ' :  x   y    . D. d ' : x   y    .

Lӡi giҧi chi tiӃt Cách 1. Lҩ\ÿLӇm M xy  d Ÿ x   y    * .

­ x' Gӑi M ' x' ; y' ÐI M WKu ® ¯ y'

2 x ­ x 2  x' œ® . 2 y ¯ y 2  y'

7KD\YjR * WDÿѭӧc   x'     y'    œ x'   y'    . Vұ\ҧnh cӫD d OjÿѭӡQJWKҷng d ' : x   y    . Cách 2. Gӑi d ' là ҧnh cӫD d TXDSKpSÿӕi xӭQJWkP I WKu d ' VRQJVRQJKRһFWUQJYӟi d QrQ SKѭѫQJWUuQK d ' FyGҥng x   y  c  . Lҩ\ N   d Jӑi N ' ÐI N WKu N ' 5;2 . LҥLFy 1  G Ÿ     F  œ F Vұ\ d ' : x   y    .

 .

Chӑn D. Ví dө 4: 7URQJ PһW SKҷng Oxy  WuP SKѭѫQJ WUuQK ÿѭӡQJ WUzQ C c là ҧnh cӫD ÿѭӡQJ WUzQ C

x 2  y 2 1 TXDSKpSÿӕi xӭQJWkP I   . A. [ 2  \  2

2

1

B. [  2  \ 2 2

1

C. [  2  \ 2 2

1

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D. [ 2  \  2

2

1


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GOODA.VN - THU VIỆN SÁCH QUÝ Lӡi giҧi chi tiӃt ĈѭӡQJWUzQ C FyWkP O   EiQNtQK R 1 . Gӑi Oc là ҧnh cӫD O TXDSKpSÿӕi xӭQJWkP I   . ­ xO  xOc ° ° 2 7DFy ® y  ° O yOc °̄ 2

xI

­x c œ® O ¯ yOc yI

2 xI  xO ­[ c œ® O 2 yI  yO ¯ \Oc

   Ÿ Oc   .   

ĈѭӡQJ WUzQ C c là ҧnh cӫD ÿѭӡQJ WUzQ C TXD SKpS ÿӕi xӭQJ WkP I   QrQ C c Fy WkP

Oc   , EiQNtQK Rc R 1 . 3KѭѫQJWUuQKÿѭӡQJWUzQ C c Oj x  2  y 2 2

1.

Chӑn C.

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