PHƯƠNG PHÁP GIẢ D

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PHNG PH?P GII TO?N. Dng 1. T?m nguy?n h?m bng phng ph?p ph?n t?ch

Phng ph?p:

t?m nguy?n h?m f(x)dx , ta ph?n t?ch

f(x) = k1.f1(x) + k2.f2(x) + ... + kn.fn(x)

Trong ?: f1(x), f2(x),...,fn(x) c? trong bng nguy?n h?m hoc ta d d?ng t?m c nguy?n h?m

Khi ?: f(x)dx = k1 f1(x)dx + k2 f2(x)dx + ... + kn fn(x)dx .

V? d 1 T?m nguy?n h?m: I = (ex + 2e-x )2dx

Li gii. 1. Ta c?: (ex + 2e-x )2 = e2x + 4 + 4.e-2x

J =

3x

+ 4.5x 7x

dx

Suy ra: I = (e2x + 4 + 4e-2x )dx = 1 e2x + 4x - 2e-2x + C 2

2. J =

3 7

x

+

4.

5 7

x

dx

=

1 ln 3

.

3 7

x

+

4 ln 5

.

5 7

x

+

C

7

7

V? d 2 T?m nguy?n h?m:

I = cos4 2xdx

J = (cos 3x.cos 4x + sin3 2x)dx

Li gii.

( ) 1. Ta c?: cos4 2x = 1 (1 + cos 4x)2 = 1 1 + 2cos 4x + cos2 4x

4

4

=

1 4

1

+

2

cos

4x

+

1

+

cos 2

8x

=

1 8

(3

+

4

cos

4x

+

cos

8x)

I

=

1 8

(3

+

4

cos

4x

+

cos

8x)dx

=

1 8

3x

+

sin

4x

+

1 8

sin

8x

+

C

2.

Ta

c?

:

cos 3x.cos 4x =

1 2

cos

7x

+

cos

x

sin3 2x = 3 sin 2x - 1 sin 6x

4

4

N?n

suy ra:

J

=

1 2

cos

7

x

+

1 cos x + 3 sin 2x -

2

4

1 4

sin

6x

dx

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= 1 sin7x + 1 sin x - 3 cos 2x + 1 cos6x + C .

14

2

8

24

Dng 2. T?m nguy?n h?m bng phng ph?p i bin s

Phng ph?p:

" Nu f (x)dx = F(x) + C th? f (u(x)).u'(x)dx = F(u (x)) + C ".

Gi s ta cn t?m h nguy?n h?m I = f (x)dx , trong ? ta c? th ph?n t?ch

f (x) = g (u(x))u'(x)dx th? ta thc hin ph?p i bin s t = u(x)

dt = u'(x)dx . Khi ?: I = g (t)dt = G(t) + C = G(u (x)) + C Ch? ?: Sau khi ta t?m c h nguy?n h?m theo t th? ta phi thay t = u(x)

V? d 3 T?m nguy?n h?m:

I

=

ln2

x x

+

1

dx

J

=

x(1

ln x.dx + 3ln x

+

2

)

K

=

ln

x3

2+ x

ln2

x

dx

Li gii.

1. t t = ln x dt = dx

x

Suy

ra

I = (t2

+

1)dt

=

t3 3

+

t

+

C

=

ln3 3

x

+ ln x + C .

2. t t = 3ln x + 2 ln x = t2 - 2 dx = 2 tdt

3

x3

Suy ra

t2 - 2 . 2 tdt

J=

33 1+ t

=

2 9

t

2

- t -1+

t

1 +

1

dt

=

2 9

t3 3

- t2 2

- t + ln(t + 1) + C

vi t = 3ln x + 2 .

3. t t = 3 ln2 x + 2 ln2 x = t3 - 2 ln xdx = 3 t2dt

x2

Suy

ra

I

=

3 2

t3dt

=

3 t4 8

+C=

3 .3 (3ln x + 2)4 8

+C

V?

d

4

T?m

nguy?n

h?m:

I

=

tan

sin4 2x.cos3 x

x

+

4

tan

x

-

4

dx

Li gii.

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Ta

c?:

tan

x

+

4

tan

x

-

4

=

tan x - 1. tan x + 1 1 + tan x 1 - tan x

=

-1

Suy ra: I = -16 sin4 x.cos6 xcos xdx

t t = sin x dt = sin xdx n?n ta c?:

I = -16 t4(1 - t2 )3dt = 16 t4(t6 - 3t4 + 3t2 - 1)dt

= 16

t11 11

-

t9 3

+

3t7 7

-

t5 5

+

C

=

16

sin11 11

x

-

sin9 3

x

+

3 sin7 7

x

-

sin5 5

x

+

C

V? d 5 T?m nguy?n h?m:

I=

tan xdx sin2 x + 3

Li gii.

t t = cos x dt = -sin xdx . Suy ra I = -

t

dt 4 - t2

? t 0 I = - t2

dt

4 t2

-

1

=

1 2

dy (vi y = 2 )

y2 - 1

t

I = 1 ln y + 2

y2 - 1 = 1 ln 2 + 2 cos x

4 cos2

x

-1

+

C

? t0

I= t2

dt = - 1 ln 2 +

4 t2

-1

2 cos x

4 cos2

x

-1

+

C

.

Dng 3. T?m nguy?n h?m bng phng ph?p tng phn

Phng ph?p:

Cho hai h?m s u v? v li?n tc tr?n a; b v? c? o h?m li?n tc tr?n a; b . Khi ? :

udv = uv - vdu ()

b

t?nh t?ch ph?n I = f (x)dx bng phng ph?p tng phn ta l?m nh sau:

a

Bc 1: Chn u,v sao cho f (x)dx = udv (ch? ?: dv = v'(x)dx ).

T?nh v = dv v? du = u'.dx .

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Bc 2: Thay v?o c?ng thc () v? t?nh vdu .

Cn phi la chn u v? dv hp l? sao cho ta d d?ng t?m c v v? t?ch ph?n vdu d t?nh hn udv .

Ta thng gp c?c dng sau

Dng

1

:

I

=

P

(x)

sin x cos x

dx

,

trong

?

P(x)

l?

a

thc

Vi

dng

n?y,

ta

t

u = P(x),

dv

=

sin x cos x

dx

.

Dng 2 : I = (x)eax+bdx

Vi

dng

n?y,

ta

t

u = P(x)

dv

=

eax+bdx

,

trong

?

P(x)

l?

a

thc

Dng 3 : I = P (x)ln (mx + n)dx

Vi dng n?y, ta t

u = ln(mx + n) dv = P(x)dx

.

Dng

4

:

I=

sin x cos x

exdx

Vi dng n?y, ta t

u

=

sin x cos x

t?nh

vdu

ta t

u

=

sin x cos x

.

dv = exdx

dv = exdx

B?I TP T LUYN

C?u 1. Nguy?n h?m ca h?m s y = (1+ sinx)2 l?:

A. 3 x 2 cos x 1 sin 2x C

2

4

B. 2 x 2 cos x 1 sin 2x C

3

4

C. 3 x 2 cos x 1 sin 2x C

2

4

D. 2 x 2 cos 2x 1 sin 2x C

3

4

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C?u 2. Nguy?n h?m ca h?m s y

3x 4

2 x 2

l?:

A.

7 x 44

1 4 x3

C

B.

4 4 x4

1 4 x3

C

C.

4 4 x4

3 4 x3

C

D.

1 4 x4

3 4 x3

C

C?u 3. Nguy?n h?m ca h?m s y 2 3x2 sin 2x l?:

A. 1 3x2 7 cos 2x 3 x sin 2x C

2

2

2

B. 1 3x2 7 cos 2x 3 x sin 2x C

2

2

2

C. 1 3x2 7 cos 2x 3 x sin 2x C

2

4

4

1 D.

3x2

7 sin 2x

3 x cos 2x

C

2

2

2

C?u 4. Nguy?n h?m ca h?m s y

sin x l?:

sin x cos x

A. 1 x ln sin x cos x C 2

B. 1 x ln sin x cos x C 2

C. 1 x ln sin x cos x C 2

D. 1 x2 ln sin x cos x C 2

C?u 5: T?m h?m s f(x) bit rng f '(x)

ax+

b x 2

,

f

'(1)

0, f (1)

4, f ( 1)

2

x2 1 5 A.

2 x2

x2 1 5 B.

2 x2

x2 1 5 C.

2 x2

D. Kt qu kh?c

C?u 6: H?m s n?o sau ?y l? mt nguy?n h?m ca h?m s f (x) Group:

x2 k vi k 0?

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