Giai tich 12 - Chuong III.doc

Nguyen ham Tch phan

Tran S TungTran S Tung

Nguyen ham Tch phan

1. Khai niem nguyen ham

( Cho ham so f xac nh tren K. Ham so F gl nguyen ham cua f tren K neu:

, (x ( K

( Neu F(x) la mot nguyen ham cua f(x) tren K th ho nguyen ham cua f(x) tren K la:

, C ( R.

( Moi ham so f(x) lien tuc tren K eu co nguyen ham tren K.

2. Tnh chat

(

(

(

3. Nguyen ham cua mot so ham so thng gap

4. Phng phap tnh nguyen ham

a) Phng phap oi bien so

Neu va co ao ham lien tuc th:

b) Phng phap tnh nguyen ham tng phan

Neu u, v la hai ham so co ao ham lien tuc tren K th:

VAN E 1: Tnh nguyen ham bang cach s dung bang nguyen ham

Bien oi bieu thc ham so e s dung c bang cac nguyen ham c ban.

Chu y: e s dung phng phap nay can phai:

Nam vng bang cac nguyen ham.

Nam vng phep tnh vi phan.Bai 1. Tm nguyen ham cua cac ham so sau:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

p) Bai 2. Tm nguyen ham F(x) cua ham so f(x) thoa ieu kien cho trc:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

Bai 3. Cho ham so g(x). Tm nguyen ham F(x) cua ham so f(x) thoa ieu kien cho trc:

a)

b)

c)

Bai 4. Chng minh F(x) la mot nguyen ham cua ham so f(x):

a)

b)

c)

d)

Bai 5. Tm ieu kien e F(x) la mot nguyen ham cua ham so f(x):

a)

b)

c)

d)

e)

f)

g)

h)

VAN E 2: Tnh nguyen ham bang phng phap oi bien so( Dang 1: Neu f(x) co dang: f(x) = th ta at .

Khi o:

= , trong o de dang tm c.

Chu y: Sau khi tnh theo t, ta phai thay lai t = u(x).

( Dang 2: Thng gap cac trng hp sau:

Bai 1. Tnh cac nguyen ham sau (oi bien so dang 1):

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

p)

q)

r)

s)

Bai 2. Tnh cac nguyen ham sau (oi bien so dang 2):

a)

b)

c)

d)

e)

f)

g)

h)

i)

VAN E 3: Tnh nguyen ham bang phng phap tnh nguyen ham tng phan

Vi P(x) la a thc cua x, ta thng gap cac dang sau:

Bai 1. Tnh cac nguyen ham sau:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

p)

q)

r)

s)


a)

b)

c)

d)

e)

f)

g)

h)

i)


a)

b)

c)

d)

e)

f)

g)

h)

i)

VAN E 4: Tnh nguyen ham bang phng phap dung nguyen ham phue xac nh nguyen ham cua ham so f(x), ta can tm mot ham g(x) sao cho nguyen ham cua cac ham so f(x) ( g(x) de xac nh hn so vi f(x). T o suy ra nguyen ham cua f(x).

Bc 1: Tm ham g(x).

Bc 2: Xac nh nguyen ham cua cac ham so f(x) ( g(x), tc la:

Bc 3: T he (*), ta suy ra la nguyen ham cua f(x).Bai 1. Tnh cac nguyen ham sau:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

VAN E 5: Tnh nguyen ham cua mot so ham so thng gap1. f(x) la ham hu t:

Neu bac cua P(x) ( bac cua Q(x) th ta thc hien phep chia a thc.

Neu bac cua P(x) < bac cua Q(x) va Q(x) co dang tch nhieu nhan t th ta phan tch f(x) thanh tong cua nhieu phan thc (bang phng phap he so bat nh).

Chang han:

2. f(x) la ham vo t

+ f(x) =

( at

+ f(x) =

( at

( f(x) la ham lng giac

Ta s dung cac phep bien oi lng giac thch hp e a ve cac nguyen ham c ban. Chang han:

+ ,

+ ,

+ ,

+ Neu th at t = cosx

+ Neu th at t = sinx

+ Neu th at t = tanx (hoac t = cotx)


a)

b)

c)

d)

e)

f)

g)

h)

i)

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a)

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m) SHAPE \* MERGEFORMAT

SHAPE \* MERGEFORMAT

1. Khai niem tch phan

( Cho ham so f lien tuc tren K va a, b ( K. Neu F la mot nguyen ham cua f tren K th:

F(b) F(a) gl tch phan cua f t a en b va k hieu la .

( oi vi bien so lay tch phan, ta co the chon bat k mot ch khac thay cho x, tc la:

( Y ngha hnh hoc: Neu ham so y = f(x) lien tuc va khong am tren oan [a; b] th dien tch S cua hnh thang cong gii han bi o th cua y = f(x), truc Ox va hai ng thang x = a, x = b la:

2. Tnh chat cua tch phan

(

(

( (k: const)

(

(

( Neu f(x) ( 0 tren [a; b] th

( Neu f(x) ( g(x) tren [a; b] th

3. Phng phap tnh tch phan

a) Phng phap oi bien so

trong o: u = u(x) co ao ham lien tuc tren K, y = f(u) lien tuc va ham hp f[u(x)] xac nh tren K, a, b ( K.

b) Phng phap tch phan tng phan

Neu u, v la hai ham so co ao ham lien tuc tren K, a, b ( K th:

Chu y: Can xem lai cac phng phap tm nguyen ham.

Trong phng phap tch phan tng phan, ta can chon sao cho de tnh

hn .

VAN E 1: Tnh tch phan bang cach s dung bang nguyen ham

Bien oi bieu thc ham so e s dung c bang cac nguyen ham c ban. Tm nguyen ham F(x) cua f(x), roi s dung trc tiep nh ngha tch phan:

Chu y: e s dung phng phap nay can phai:

Nam vng bang cac nguyen ham.

Nam vng phep tnh vi phan.

Bai 6. Tnh cac tch phan sau:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

VAN E 2: Tnh tch phan bang phng phap oi bien so

Dang 1: Gia s ta can tnh .

Neu viet c g(x) di dang: th

Dang 2: Gia s ta can tnh .

at x = x(t) (t ( K) va a, b ( K thoa man ( = x(a), ( = x(b)

th

Dang 2 thng gap cac trng hp sau:


Bai 3. Tnh cac tch phan sau (oi bien so dang 1):

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

p)

Bai 4. Tnh cac tch phan sau (oi bien so dang 2):

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

VAN E 3: Tnh tch phan bang phng phap tch phan tng phan

Vi P(x) la a thc cua x, ta thng gap cac dang sau:



a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

o)

p)

q)

VAN E 4: Tnh tch phan cac ham so co cha gia tr tuyet oie tnh tch phan cua ham so f(x) co cha dau GTT, ta can xet dau f(x) roi s dung cong thc phan oan e tnh tch phan tren tng oan nho.Bai 2. Tnh cac tch phan sau:

a)

b)

c)

d)

e)

f)

g)

h)

i)


a)

b)

c)

d)

e)

f)

g)

h)

i)

VAN E 5: Tnh tch phan cac ham so hu t

Xem lai cach tm nguyen ham cua cac ham so hu t.


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

VAN E 6: Tnh tch phan cac ham so vo t

Xem lai cach tm nguyen ham cua cac ham so vo t.


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

p)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)

g)

h) i)


a)

b)

c)

d)

e)

f)

g)

h)

i)

VAN E 7: Tnh tch phan cac ham so lng giac

Xem lai cach tm nguyen ham cua cac ham so lng giac.


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

p)

q)

r)

s)


a)

b)

c)

d)

e)

f)

g)

h)

i)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

p)

VAN E 8: Tnh tch phan cac ham so mu va logarit

S dung cac phep toan ve luy tha va logarit. Xem lai cac phng phap tm nguyen ham.


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

VAN E 9: Mot so tch phan ac bietDang 1. Tch phan cua ham so chan, ham so le

( Neu ham so f(x) lien tuc va la ham so le tren [-a; a] th

( Neu ham so f(x) lien tuc va la ham so chan tren [-a; a] th

V cac tnh chat nay khong co trong phan ly thuyet cua SGK nen khi tnh cac tch phan co dang nay ta co the chng minh nh sau:

Bc 1: Phan tch

Bc 2: Tnh tch phan bang phng phap oi bien. at t = x.

Neu f(x) la ham so le th J = K( I = J + K = 0

Neu f(x) la ham so chan th J = K ( I = J + K = 2K

Dang 2. Neu f(x) lien tuc va la ham chan tren R th:

(vi ( ( R+ va a > 0)

e chng minh tnh chat nay, ta cung lam tng t nh tren.

e tnh J ta cung at: t = x.

Dang 3. Neu f(x) lien tuc tren th

e chng minh tnh chat nay ta at:

Dang 4. Neu f(x) lien tuc va hoac

th at: t = a + b x

ac biet,neu a + b = (

th at

t = ( x

neu a + b = 2(th at

t = 2( x

Dang 5. Tnh tch phan bang cach s dung nguyen ham phu

e xac nh nguyen ham cua ham so f(x) ta can tm mot ham g(x) sao cho nguyen ham cua cac ham so f(x) ( g(x) de xac nh hn so vi f(x). T o suy ra nguyen ham cua f(x). Ta thc hien cac bc nh sau:

Bc 1: Tm ham g(x).

Bc 2: Xac nh nguyen ham cua cac ham so f(x) ( g(x), tc la:

Bc 3: T he (*), ta suy ra la nguyen ham cua f(x).Bai 1. Tnh cac tch phan sau (dang 1):

a)

b)

c)

d)

e)

f)

g)

h)

i)

Bai 2. Tnh cac tch phan sau (dang 2):

a)

b)

c)

d)

e)

f)

g)

h)

i)


a) (n ( N*)b)

c)

d)

e)

f)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

n)

o)

VAN E 10: Thiet lap cong thc truy hoiGia s can tnh tch phan (n ( N) phu thuoc vao so nguyen dng n. Ta thng gap mot so yeu cau sau:

( Thiet lap mot cong thc truy hoi, tc la bieu dien In theo cac In-k (1 ( k ( n).

( Chng minh mot cong thc truy hoi cho trc.

( Tnh mot gia tr cu the nao o.Bai 1. Lap cong thc truy hoi cho cac tch phan sau:

a) ( at

b) ( at

c)

( Phan tch:

d)

( at

( at

e)

( at

f)

( at

g)

( at

(at

h)

( Phan tch

Tnh .at

i)

( at

k)

( Phan tch ( at


1. Dien tch hnh phang

( Dien tch S cua hnh phang gii han bi cac ng:

o th (C) cua ham so y = f(x) lien tuc tren oan [a; b].

Truc hoanh.

Hai ng thang x = a, x = b.

la:

(1)


o th cua cac ham so y = f(x), y = g(x) lien tuc tren oan [a; b].

Hai ng thang x = a, x = b.

la:

(2)

Chu y:

( Neu tren oan [a; b], ham so f(x) khong oi dau th:

( Trong cac cong thc tnh dien tch tren, can kh dau gia tr tuyet oi cua ham so di dau tch phan. Ta co the lam nh sau:

Bc 1: Giai phng trnh: f(x) = 0 hoac f(x) g(x) = 0 tren oan [a; b]. Gia s tm

c 2 nghiem c, d (c < d).

Bc 2: S dung cong thc phan oan:

=

(v tren cac oan [a; c], [c; d], [d; b] ham so f(x) khong oi dau)


o th cua x = g(y), x = h(y)(g va h la hai ham so lien tuc tren oan [c; d])

Hai ng thang x = c, x = d.

2. The tch vat the

( Goi B la phan vat the gii han bi hai mat phang vuong goc vi truc Ox tai cac iem cac iem a va b.

S(x) la dien tch thiet dien cua vat the b cat bi mat phang vuong goc vi truc Ox tai iem co hoanh o x (a ( x ( b). Gia s S(x) lien tuc tren oan [a; b].

The tch cua B la:

( The tch cua khoi tron xoay:

The tch cua khoi tron xoay do hnh phang gii han bi cac ng:

(C): y = f(x), truc hoanh, x = a, x = b (a < b)

sinh ra khi quay quanh truc Ox:

Chu y: The tch cua khoi tron xoay sinh ra do hnh phang gii han bi cac ng sau quay xung quanh truc Oy:

(C): x = g(y), truc tung, y = c, y = d

la:

VAN E 1: Tnh dien tch hnh phangBai 10. Tnh dien tch hnh phang gii han bi cac ng sau:

a)

b)

c)

d)

e)

f)

g)

h)

Bai 11. Tnh dien tch hnh phang gii han bi cac ng sau:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)


a)

b)

c)

d)

e)

f)

g)

h)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)


a) , tiem can xien cua (C), x = 1 va x = 3.

b) , tiem can xien cua (C), x = 1 va x = 2

c) va tiep tuyen vi (C) tai iem co hoanh o x = 2.

d) va tiep tuyen ci (C) tai iem co hoanh o x = 2.

e) va cac tiep tuyen vi (C) tai O(0 ; 0) va A(3; 3) tren (C).

VAN E 2: Tnh the tch vat theBai 1. Tnh the tch vat the tron xoay sinh ra bi hnh (H) gii han bi cac ng sau quay quanh truc Ox:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

Bai 2. Tnh the tch vat the tron xoay sinh ra bi hnh (H) gii han bi cac ng sau quay quanh truc Oy:

a)

b)

c)

d)

Bai 3. Tnh the tch vat the tron xoay sinh ra bi hnh (H) gii han bi cac ng sau quay quanh: i) truc Ox

ii) truc Oy

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)



a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

o)

p)

q)

r)

s)

t)


a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

o)

p)

q)

r)

s)

t) Bai 4. Tnh cac tch phan sau:

a)

b)

c)

d)

e)

f)

g)

h)

i)

k)

l)

m)

o)

p)

q)

r)

s)

t)

Bai 5. Tnh dien tch cac hnh phang gii han bi cac ng sau:

a)

b)

c)

d)

e)

f)

g)

h)

m)

n)

o) , tiep tuyen tai giao iem cua (C) vi truc tung.

p) , tiep tuyen tai iem M thuoc o th co hoanh o x = .

Bai 6. Tnh the tch cac vat the tron xoay sinh ra khi quay hnh phang gii han bi cac ng sau quanh truc:

a)

b)

c)

d)

e)

f)

Chan thanh cam n cac ban ong nghiep va cac em hoc sinh a oc tap tai lieu nay.

[email protected] III

NGUYEN HAM, TCH PHAN VA NG DUNG

I. NGUYEN HAM

( EMBED Equation.DSMT4




( EMBED Equation.DSMT4 ( EMBED Equation.DSMT4







f(x) co chaCach oi bien EMBED Equation.DSMT4 EMBED Equation.DSMT4

hoac EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4

hoac EMBED Equation.DSMT4

EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 uP(x)P(x)P(x)lnxdv EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 P(x)

CHNG III

NGUYEN HAM, TCH PHAN VA NG DUNG

II. TCH PHAN

f(x) co chaCach oi bien EMBED Equation.DSMT4 EMBED Equation.DSMT4



hoac EMBED Equation.DSMT4

EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 uP(x)P(x)P(x)lnxdv EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 P(x)

III. NG DUNG TCH PHAN

IV. ON TAP TCH PHAN

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Giai tich 12 - Chuong III.doc

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