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Nguyen ham Tch phan Tran S Tung
Trang 78
1. Khai niem nguyen ham Cho ham sofxac nh tren K. Ham soFgl nguyen ham cuaftren K neu:
'( ) ( )F x f x= , "x K Neu F(x) la mot nguyen ham cuaf(x) tren K th ho nguyen ham cuaf(x) tren K la:
( ) ( )f x dx F x C = + , C R.
Moi ham sof(x) lien tuc tren K eu co nguyen ham tren K.2. Tnh chat
'( ) ( )f x dx f x C = + [ ]( ) ( ) ( ) ( )f x g x dx f x dx g x dx =
( ) ( ) ( 0)kf x dx k f x dx k =
3. Nguyen ham cua mot so ham so thng gap
4. Phng phap tnh nguyen hama) Phng phap oi bien so
Neu ( ) ( )f u du F u C = + va ( )u u x= co ao ham lien tuc th:
[ ] [ ]( ) . '( ) ( )f u x u x dx F u x C = +
b) Phng phap tnh nguyen ham tng phanNeu u, v la hai ham so co ao ham lien tuc tren K th:
udv uv vdu= -
CHNG IIINGUYEN HAM, TCH PHAN VA NG DUNG
I. NGUYEN HAM
0dx C=
dx x C = +
1
, ( 1)1xx dx C
+
= + -+
a
a aa
1 lndx x C x
= +
x xe dx e C = +
(0 1)ln
xx aa dx C a
a= + <
cos sinxdx x C = +
sin cosxdx x C = - +
2
1 tancos
dx x C x
= +
2
1 cotsin
dx x C x
= - +
1cos( ) sin( ) ( 0)ax b dx ax b C aa
+ = + +
1sin( ) cos( ) ( 0)ax b dx ax b C aa
+ = - + +
1 , ( 0)ax b ax be dx e C aa
+ += +
1 1 lndx ax b C ax b a
= + ++
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Nguyen ham Tch phan Tran S Tung
Trang 80
Bai 5. Tm ieu kien e F(x) la mot nguyen ham cua ham sof(x):a)
3 2
2( ) (3 2) 4 3. .( ) 3 10 4
F x mx m x xTm m
f x x x
= + + - +
= + -b)
2
2
( ) ln 5. .2 3( )
3 5
F x x mxTm mx
f xx x
= - ++
=+ +
c)
2 2
2
( ) ( ) 4. , , .( ) ( 2) 4
F x ax bx c x xTm a b c
f x x x x
= + + - = - -
d)
2( ) ( ). , , .( ) ( 3)
x
x
F x ax bx c eTm a b cf x x e
= + + = -
e)2 2
2 2( ) ( ) . , , .( ) (2 8 7)
x
x
F x ax bx c eTm a b c
f x x x e
-
-
= + +
= - - +f)
2
2( ) ( ) . , , .( ) ( 3 2)
x
x
F x ax bx c eTm a b c
f x x x e
-
-
= + +
= - +
g) ( ) ( 1)sin sin 2 sin3 . , , .2 3( ) cos
b cF x a x x x Tm a b cf x x
= + + + =
h)
2
2( ) ( ) 2 3
. , , .20 30 7( ) 2 3
F x ax bx c xTm a b cx x
f xx
= + + -
- += -
VAN E 2: Tnh nguyen ham ( )f x dx bang phng phap oi bien so
Dang 1: Neu f(x) co dang: f(x) = [ ]( ) . '( )g u x u x th ta at ( ) '( )t u x dt u x dx= = .
Khi o: ( )f x dx = ( )g t dt , trong o ( )g t dt de dang tm c.
Chu y: Sau khi tnh ( )g t dt 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) (5 1)x dx- b) 5(3 2 )
dx
x- c) 5 2xdx-
d) 2 7(2 1)x xdx+ e)3 4 2( 5)x x dx+ f) 2 5
xdx
x +
g) 2 1.x xdx+ h)2
3
3
5 2
xdx
x+ i) 2(1 )
dx
x x+
k) 4sin cosx xdx l) 5sincosx dxx
m) 2tancosxdx
x
f(x) co cha Cach oi bien
2 2a x- sin ,
2 2x a t t = -
p p
hoac cos , 0x a t t = p
2 2a x+ tan ,
2 2x a t t = - < = - + =
g) 2 ,y x y x= = h) ( )2 2 4 1x y+ =
i) 149
22
=+yx
k) 1, 2, 0, 0y x y y x= - = = =
l) 2 0, 2, 0x y y x- = = = m) 2 3, 0, 1y x y x= = =
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Tran S Tung Nguyen ham Tch phan
r)/2
0
cos3sin 1
xdx
x
p
+s)
/2
2 20
sin
sin 2 cos cos2
xdx
xx x
p
+ t)
/3 2
20
sin
sin 2 cos
x xdx
x x
p
Bai 4. Tnh cac tch phan sau:
a)
32
0 ln( 5)x x dx+ b) -
3
2
2
)ln( dxxx c)
12
0 ( 2)x
x e dx-
d)/2
sin
0( cos ) cosxe x x dx
p
+ e)ln 5
ln 3 2 3x x
dx
e e-+ - f)
2 2
1ln
e
x x dx
g)3
1
1 lnex
xdxx
+ h)
12
0( 1) xx e dx+ i)
1
0 1x
dx
e+
k)2 2
20 ( 2)
xx edx
x + l)
12 2
0(4 2 1) xx x e dx- - m)
2
21
ln(1 )xdx
x
+
o) /2 3
0sin5xe x dxp p) 2
1
lne x dxx
q)1 2
0ln(1 )x x dx+
r)1
3 2 ln
1 2 ln
e xdx
x x
-
+ s)
+e
dxx
xx
1
ln.ln31 t)3 2
1
ln
ln 1
e xdx
x x +
Bai 5. Tnh dien tch cac hnh phang gii han bi cac ng sau:a) 3 3 1, 0, 0, 1y x x y x x= - + = = = - b)
4, 0, 2, 1
2y y x x
x= = = - =
-
c) 4 21 92 , 04 4
y x x y= - + + = d) , 2, 1xy e y x= = =
e) 1 11 , 0, 2, 42 1
y x y x xx
= - + = = =-
f) 2 22 , 4y x x y x x= - = - +
g)2 1
, 0, 01
xy y x
x
+= = =
+h)
2, 0
1x x
y yx
- += =
+
m)2 3 2 , , 0, 1
1x x
y tiem can xien x xx
+ -= = =
+
n)2 2 , 0,
1x x
y y tiep tuyen ve t goc toa ox
+ -= =
+
o) 3 23 3 1y x x x= + + + , tiep tuyen tai giao iem cua (C) vi truc tung.
p) 31 34
y x x= - , tiep tuyen tai iem M thuoc o th co hoanh o x = 2 3 .
Bai 6. Tnh the tch cac vat the tron xoay sinh ra khi quay hnh phang gii han bi cacng sau quanh truc:a) , 0, 3;y x y x Ox= = = b) ln , 0, 1, ;y x x y x x e Ox= = = =
c) , 0, 1;xy xe y x Ox= = = d) 2 24 , 2;y x y x Ox= - = +
e) 2 4 , 0;y x x Oy= - = f) , 0, 1;yx ye x y Oy= = =
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