[r]
(1)NGUYÊN HÀM VÀ TÍCH PHÂN Hai nguyên hàm đặc biệt:
2 2
2
1/ ;2/
2 2
dx x k
ln x x k C x kdx x k ln x x k C
x k
I.Nguyên hàm hàm số lượng giác:
1/
5
4 (1 )
5
sin x sin x sin xcos xdx sin x sin x dsinx C
2/
9 11
5 2
(1 ) cos
9 11
cos x cos x cos x sin xcos xdx cos x cos xd x C
3/
2
4 2 .1 (1 2 )(1 22 )
2
cos x cos x
sin xcos xdx dx cos x cos x dx
3
1 6
1 2
8 16 2 32 2
cos x cos x cos x sin x sin x sin x
cos x cos x dx cos x dx x
4/ 8 (3cos ) 3( ) 11
4
cos xsin xdx sin x x cos x dx sin x sin x sin x sin x dx
1 11
8 11
cos x cos x cos x cos x C
.
5/ cos2 1 1ln ln tan( / 2)
sin cos cos
dx d x cosx
dx C x C
x cos x x x cosx
6/ 2sin (3 ) (3 ) 17 (3 )
3 4 25 25
x cosx a sinx cosx b cosx sinx d sinx cosx
dx dx dx
sinx cosx sinx cosx sinx cosx
6
ln
25x sinx cosx C
.
7/ cos ( ) ( ) 1 ln
1 2
dx xdx sinx cosx sinx cosx
dx x sinx cosx C
tanx sinx cosx sinx cosx
.
8/ 3 ( ) ( 3 ) (2 / 4) ( ) ( )
( ) ( ) ( / 4)
sinx sinx cosx sinx cosx d x
dx dx sinx cosx d sinx cosx
sinx cosx sinx cosx cos x
2
( / 4) ( )
tan x sinx cosx C
.
9/ 2
( / 2)
3 ( / 2) 10 ( / 2) ( / 2) ( / 2)(3 ( / 2))
dx dx d x
sinx cosx cos x sin x cos x cos x tan x
( / 2)
ln ( / 2) ( / 2)
dtan x
tan x C
tan x
(2)10/ 2
( / 2)
7 16 ( / 2) 12 ( / 2) ( / 2) ( / 2) ( / 2)
dx dx dtan x
cosx sinx cos x sin x cos sin x tan x tan x
1 1 ( / 2)
( / 2) ln
2 ( / 2) ( / 2) 2 ( / 2) tan x
dtan x C
tan x tan x tan x
.
11/
2
2
4 2 3
sin x sin x dx tan x
dx tan xdtanx C
cos x cos x cos x
.
12/ tan xdx3 (tan x tanx tanx dx3 ) tanx tan x( 1)dx sinxdx cosx/ (tan x2 ) / ln cosx C . 13/ cot xdx4 (cot x cot x cot x4 1 1)dxcot x cot x2 ( 1)dx (cot x2 1)dxdx
3
(cotx) / cotx x C
.
14/ 2 ( / /8)
2 2 ( / 4) 2 ( / /8)
dx dx dx cot x
C
sinx cosx cos x sin x
.
15/
3
3
2 / /
3
3
3 sin x sinx sin x sinx cotx
cotxdx dx cot xcotxdcotx cot xdcotx cot x C
sin x sin x sin x
.
16/
4
3/ 4
2
3
4
dx tanxdx
tan xdtanx tanx C tanxcos x
sin xcos x
.
17/ 3 ( ) 1/ ( ) 3( )2 /
2 sinx cosx
dx sinx cosx d sinx cosx sinx cosx C sinx cosx
.
18/
2
3 3
( 2)
2
( 2)
( 2) ( 2)
d t
cos xdx dsin x
t sinx cosx
sinx cosx sinx cosx t
3
2 1
t
dt C
t t t
.
19/
2
1
( )
4
2 6 8 2 3/ 4
cosxdx dsinx dsinx u
C sinx sinu
cos x sin x sin x
.
20/ sin x22 dx dx2 2sinxcosx2 dx tanx 2lncosx C cos x cos x cos x
.
21/
2
3
3 3
(1 )
( 3 ) 3ln
4 2
dx dx tan x dtanx t t
t t t t dt t C
sin xcos x cos xtan x tan x t
.
22/ 2 2 2 2 ( 2 ) 1
3 4 3 2
sinx cosx dcosx dsinx d tant
dx dsinx
sin x cos x cos x sin x tan t sinx sinx
2
ln ( )
2 sinx
t C cosx tant sinx
.
23/ cosx cos xdx x ( 0; / ) cosxdcosx2cos x3 / / 3C 24/
2 1 1 2( / 3) 1 1
2
2 ( / 3)
3
cos xdx cos xdx cos t
dt cost sint dt
sin x sint sint
sinx cosx
1
ln ( / 3)
8
t
tan sint cost C t x
(3)25/
2 2 2
2
2
2 2
sin cos /( )
( )
x xdx tdt b a a cos x b sin x
a b C
t b a
a cos x b sin x
.
26/ ( ) 2 2 ln ( ) ( )
4
2 ( )
cosx sinx d sinx cosx dtant dt t
dx tan C tant sinx cosx
cost
sin x sinx cosx tan t
.
27/ 2 ( (0; / 4)) ( ) 2 ln
2 cos x
tan x cot x dx x cotx tanx dx dx sin x C
sin x
.
28/ 2 2 2 2
3
4
sin xdx sinxdsinx
d sin x sin x C
cos x sin x sin x
.
29/ ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
sin x a x b dx dx
a b k
cos x a cos x b sin a b cos x a cos x b
1 ( ) ( ) ( )
( ) ( ) ln
( ) ( ) ( ) ( ) ( )
sin x a sin x b cos x b
d x a d x b C
sin a b cos x a cos x b sin a b cos x a
.
30/ ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
cos x a x b dx dx
a b k
sin x a cos x b cos a b sin x a cos x b
1 ( ) ( ) ( )
( ) ( ) ln
( ) ( ) ( ) ( ) ( )
cos x a sin x b sin x a
d x a d x b C
cos a b sin x a cos x b cos a b cos x b
.
BÀI TẬP : Tìm nguyên hàm sau:
3
2
cos 4
1/ ;2 / ;3/ ;4 / ;5/ ;6 / ;7 / ;
3 1
dx xdx sin xdx dx sin xsin x cos xdx sin xdx
dx
cosx cosx sinx cos x cosx tanx cot x cos x cosx cosx
3
2
1 2
8 / ;9 / ;10 / ;11/ ;12 / ;13/ ;
2
cos xdx sinx sin x sinxdx dx dx sin x cos x
dx dx
sin x sinx cos x sin x sin x cos x sinx cosx
4
2
4
14 / ;15/ ;16 / ;17 / ;18/ ;19 / ;
1 2 2
dx sin x dx tan x dx dx
dx dx
sin xcos x cosx sinxcos x cos x sin x sinx cosx
20 / ;21/ ;22 / ;23/ 61 sin
( ) ( )
1 3
sin x sinx sinx cosx dx
dx dx cos x xcos x
cos x a cos x b
cosx sin x
.
24 / ;25/ ( ) ;26 / ;27 /
2
dx sinxdx sinxcosxdx
tanxtan a x dx
sinx cosx
sinx cosx sin x
II.Nguyên hàm hàm phân thức hữu tỉ:
1/ ln
( 1)(2 1) 1 2
xdx a b dx dx x
dx C
x x x x x x x
.
2/
2
2 2
1 1
ln
( 2) ( 2) 4
dx dx x
dt C
x x x x t t x
(4)3/
10
10 10 10 2 2
1 ( 1) 1
( 1) 10 ( 1) 10 ( 1) 10 ( 1) 10 ( 1)
dx dx dt t t
dt dt
x x x x t t t t t t t
10
2 10 10
1 1 1 1 1
ln ln
10 10 10 1
t x
dt C C
t t t t t x x
.
4/
2
4 2
1 1/ ( 1/ ) 1 1
ln
1 1/ ( 1/ ) 2 2 2 2
x x d x x t
dx dx dt C
x x x x x t t t
.
5/
2
2 2 2 2 4 3
( ) / 1 2
( 0)
( ) ( ) / 2
dx d atant adt cos t cos t sin t
a dt t C
x a a tan t a a cos t a a
.
6/
2
10 10
1 4 1
( )
(2 ) 27 27
x dx t t
dt C t x
x t t t t
.
7/
4 2
3
6 6
1 1
( ) /
1 1
x x x x dx dx
dx dx arctanx arctanx C
x x x x
.
8/
3 2
6 3 2
( ) (1 1/ ) ( 1/ )
ln
4 1/ 4( 1/ ) ( 1/ ) ( 1/ ) ( 1)
x x dx x dx d x x dt t
C
x x x x x x x x x x x t t t
.
BÀI TẬP: Tìm nguyên hàm sau:
3 2 2001
9 2 2 1002
1 ( 1)
1/ ;2 / ;3/ ;4 / ;5/ ;6 /
3 ( ) ( 1) ( 1)( 1) (1 )
dx x dx x x dx x dx x dx
dx
x x x k x x x x x x x x
.
III Nguyên hàm hàm số có chứa ẩn dấu thức:
1/
3
2
3
1 ( 1) / 1
( ) ( 1)
3
3
x t t
dx t dt t t dt t C t x
t x
.
2/
2
( 1) 1
( 1) ( 1)
2(1 ) 2
1
xdx t tdt t t
t tdt C t x
t x
.
3/
3
2
( 2)
2 ( 2)
3
x dx t tdt t
t C t x t
x
.
4/
3 2
4
3
(3 ) / 1 3
1 ln ( 1)
1 4
1
x dx t dt t t
dt t dt t t C t x
t t t
x
.
5/ 1 / /
( 1) ( 1)
2
1
dx x x
dx x x C
x x
(5)6/
6 3
2
3
3
1
6 6 ln ( )
1
dx dt t dt t t
t t dt t t C t x
t t t t
x x
.
7/
2
2
1 1 1 1
ln ln
( 1) 1 2
1 1
dx tdt t x
dt C C
t t t t t
x x x
.
8/ 2 2 2
( 1) 1 1
ln
( 1) 1
( 1) 2 ( 1) 2
dx x dx tdt t
dt C
t t t t t
x x x x x x
.
9/ 2
1
dx dsinx cosxdx
sinx cosx sinx cosx
x x
10/ 2 2 3 2 2 2 3 2
( ) ( )
dx datant costdt sint
C
a a
a x a a tan t
.
11/
2
2
6
1
(1 )
3
x costdsint cot t cot t
dx cot t cot t dcott C
x sin t
.
12/
2
(1/ )
2 /
1 2
2
dx d t dt
t C x C
t x x x
t t t
.
13/
2
4 /( 1)
(1/ 1)
2
1 2
( 1) 3 2 1
x
dx d t dt t
C C
t
x x x
t t t t
.
14 1 22 .4 2 2 8 2 2 ( 4 2 2 1)
1
1
x cos t
dx dcos t tant cos t sin tdt sin tcos tdt cos t cos t dt cos t
x
0,5sin t4 2sin t2 2t C
.
BÀI TẬP: Tìm nguyên hàm sau:
2
2
2 /
2
1
1/ ;2 / ;3/ ;4 / ;5/ ;5'/
(1 )
2 1 2
dx dx xdx x dx x x
x x dx dx
x
x x x x x x x
3
2
2 2 2
6 / ;7 / ;8/ ;9 / (1 ) ;10 / ;11/
1 (2 1) ( 1) (1 n) 1n n
x dx dx dx x dx dx
x dx
x x x x x x x x x
.
IV Nguyên hàm hàm số mũ hàm số Lơgarít:
1/ 1 1ln
2 ( 2) 2 2
x x
x
x x x x x x
dx de e
de C
e e e e e e
(6)2/ ( 3) / ( 3) 2( 3) 1/ 2 /
( 3)
x
x x x x
x
e dx
e d e e C e C
e
.
3/
2 1
1 ln( 1)
1 1
x x x
x x x
x x x
e dx e de
de e e C
e e e
.
4/
2
2
2 2 2
1 1 1
ln
3 ( 3) 6
x x
x
x x x x x x
dx de e
de C
e e e e e e
.
5/ 2
2 2
2 ( )
( 2) 2
( 1)
x
x
x x
e dx tdt d tanu
du u C e tanu
t t tan u
e e
.
6/
2
3 1 2
( 1) ( 1)
3 3
x t t t t x
e dx e d te dt e t C e x C
.
7/
2 ln ( / / 2) ( )
1
x x
x
x x x
e dx e dx dtant dt
tan t C t arctane cost
e e e tan t
.
8/ 1 22 21 2( ) ( )
1
x x
x x
x
e e dx t tdt
e dx dt t u C t e tanu
e t t
.
9/
2
2
(1 )
( )
1
x x
x
x x
e e dx
dx dx x t C t arctane
e e
.
10/
2
4
ln 3ln 1 2
( ) ( 3ln 1)
3 9
x x t t t
dx t tdt t t dt C t x
x
.
11/ (2ln ) 2 1 2ln 2ln
1 ( 1) 1
1
x
x x
dx d t dt t e
dt C C
t t t t t t
e e
.
V Tìm ngun hàm theo phương pháp tính phần:
1/ Các nguyên hàm dạng: xdf x( ), ví dụ: 2
2
; xdx ; ; xsinx
xsin dx xtan xdx dx
sin x cos x
.
2/ Các nguyên hàm dạng: f x dx( ) , ví dụ: ln2xdx; ln(x x2 k dx) ; ln (2 x x2 k dx) ;
3/ sinxln(1cosx dx) ln(1cosx d) (1cosx) (1 cosx) ln(1 cosx) C. 4/ cosxln(1cosx dx) ln(1cosx dsinx sinx) ln(1cosx) x sinx C .
5/ F sin(ln )x dx xsin (ln )x cos(ln )x dx xsin (ln )x xcos(ln )x F F x sin (ln )x cos(ln ) / 2x C.
6/
2 2
2
1 1 1
ln ln ln ln
1 1 2
x x x x x x x
x dx dx x C
x x x x x
(7)7/
2
1/
ln( ) ln( ) cos x ln( ) ln ( / 2)
sinx tanx dx cosx tanx cosx dx cosx tanx tan x C tanx
.
8/ e cosxdx e sinx cosxx x( ) / C e sinxdx e sinx cosx; x x( ) / C
.
9/
2 ( 1)
( 1) 1
x x x
x x
xe xe e
dx xe d x e dx C
x x x
.
10/
2
2 2
2
ln( )
ln( ) ln( )
x x x k
dx x x k d x k x k x x k x C
x k
.
11/ 2 2
1
ln ln ln ln ln ln ln ln
dx dx dx x dx x
dx C
x x x x x x x x
.
12/
1
( / 2) ( / 2) ( / 2) ( / 2)
1 ( / 2)
x
x x x x x
sinx e dx
e dx e tan x dx e dtan x tan x de e tan x C
cosx cos x
.
BÀI TẬP: Tìm nguyên hàm sau:
2
3 2
2 2
ln( )
1/ ;2 / (ln ) ;3/ ;4 / ;5/ ;6 / ;7 /
(1 ) ( 2)
x
x x
xcosxdx sinx x sinx x e dx
cos x dx x e dx e sin xdx dx dx
sinx sin x cosx x
.
VI Một số tích phân đặc biệt:
1/ Nếu f(x) hàm liên tục đoạn 0; thì:
0
( ) ( )
2
xf sinx dx f sinx dx
.
a/
0
0
( 2) ln
ln( 2)
2 2 2
xsinxdx d cosx
cosx
cosx cosx
.
b/
1
1
2 2
1
01 10 11 2 4
xsinxdx dcosx dt
arctant
cos x cos x t
.
2/ Nếu f(x) hàm liên tục đoạn 0;1 thì:
/ /
0
( ) ( )
f sinx dx f cosx dx
.
Áp dụng:
/ / / /
0 0
1 ( )
;
2
n n n n n
n n n n n n n n
sin xdx cos xdx sin x cos x dx sinxdx
sin x cos x sin x cos x sin x cos x sinx cosx
(8)/ / /
0 0
; ( ) ( )
4
n
n n
n n
n n
cosxdx
sinx cosx dx sin x cos x dx sinx cosx
.
3/ Nếu f(x) hàm liên tục lẻ đoạn a a; thì: ( )
a
a
f x dx
.
Áp dụng:
/
2 2 2
/
ln ( 1) 0; 0; 0;
a a
n m n m n m
a a
x x x dx tan x x dx x cos xdx
1/ / /
1
2
1/ / /
1
ln 0; 0;
1
x x
n m m
x x
x e e x x x x dx
cos xdx cos xdx dx
x e e cos x cos x
.
4/ Nếu f(x) hàm liên tục chẵn đoạn a a; thì:
0
( )
( )
a a
x a
f x dx
f x dx b
Áp dụng: a/
/ / /
3
/ 0
(1 ) / 2x
cos xdx
cos xdx sin x dsinx
.
b/
1
1
1
( ) ( ) 1/
1
x x
x x x x
x o
e e
dx e e dx e e e e
.
c/
/ /
4 2
/ 0
(1 ) 1/ / 1/ 8/ 315 x
sin xcos x
dx sin xcos xdx t t dt e
.
5/ Nếu f(x) hàm liên tục tuần hồn với chu kì T > thì:
0
( ) ( )
a T T
a
f x dx f x dx
.
Áp dụng:
200 2
0 0
1 cosxdx 100 cosxdx 100 cos x( / 2)dx 200 cost dt
/
0 /
200 costdt costdt 200 2(1 1) 400
.
VII.Một số toán lẻ:
2 /
/
2
0 0
(1 )
1/ ;2 / ;3/ ( ) ;4 / ;5/
1 1
p p
x dx x ln x x sinx
dx ln tanx dx dx dx
x x x cosx
(9)