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

 !"#
$%&#'()*+#
$,- ./'()*+#
$0 ''()*+# 
$12'3/,45
* ,,456'47'4489:;<+47'44='4!
$>'3/?@<+#A:;'BC'./,45DE'FG
H3I'!+JE
$,'()*+#./#A:;+#:;8' KLMBC',-<+ ''()*+#N19
:;<+'()*+#='4!O
$,,45./#A:;+#:;8' KLMBC',-<+ ''()*+#N19
:;<+,45='4!='4!O
$P4BC',458Q,B&,F4R'
$P4BC',458Q,Q,<SQ
<-8TCQ
UV()!"#W
X%&#'()*+#W
$12'3/W !
"#$%&$%'()
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a/-(.&$%/0) !
1234.* !564)7!*$%&$%/
XH,- ./'()*+#
$12YVW3"8!95
!2
[ ]
K O K O Z[ \KZOBZ] 'KZOBZf x g x d+
∫ ∫ ∫
12

 \KZOBZ[ \KZOBZ
∫ ∫
'(:;;<
X^0 ''()*+# 

 _BZ[
∫

 BZ[ BZ x C= +
∫ ∫
H
( )


 Z BZ[ 
]
x C
α α
α
α
+
+ ≠ −
∫
^

 BZ[Y Z
Z
C+
∫
`abY+6':;_/c

a/

 :ZBZ[d E:Z]

c
∫
b/

 E:ZBZ[ :Z]

∫
c/
Z Z

 e BZ[ e ]

∫
d/
Z Z

 / BZ[ / ] K_ O
Y/
a< ≠
∫

f
1
a/
H


 BZ[/Z]
E: x
∫
b/
H

 BZ[dEZ]
: x
∫
X`,45
X`X12'3/,45WE+#:;\Y*CD*%<+/NY+/:;-F(A%X(g
Y+#A'()*+#.\D*%F&(:;
gKOhgK/O
87i'GY+,45./\=/8<+,&(Y+
K O Z
b
a
f x d
∫
jVW
$(/k/'G
K O Z
b
a
f x d
∫
Y+,45./\D*8EJl/mn
$
K O Z
b

a
f x d
∫
=
K O
b
a
F x
= F(b) – F(a)
* ,45o'4C(A<+EY-),45
K O Z
b
a
f x d
∫
=
K O
b
a
f t dt
∫
=
K O
b
a
f v dv
∫
=
X`XH,-./,45


K O Z _
a
a
f x d =
∫
H
K O Z K O Z
b a
a b
f x d f x d= −
∫ ∫
^
K O Z K O Z] K O Z
b c b
a a c
f x d f x d f x d=
∫ ∫ ∫
`
l K O K On Z K O Z] K O Z
b b b
a a a
f x g x d f x d f x d+ =
∫ ∫ ∫
f
K O Z K O Z
b b
a a
kf x d k f x d=
∫ ∫
2/ BJ'EDE'7'

HXX,'()*+#6':MBC',-<+'()*+# 
Ví dụ 1W,'()*+#./+#:;:/(
/f(x) = x
3
+
^
 H
x x
+
 f(x) =
K OK Ox x x+ − +
f(x) =
^ H
H
x x
x
− −
Bf(x) = x
4
+ 3x
3
-5x + 2
Ví dụ 2W,'()*+#./+#:;:/(
/f(x) =
H
H E:HZ]H:^Z]Zc
\KZO[`:
H
Z
\KZO[

^
p f
 
K OK Ox x
x x
− −
Bf(x) =
H

H: Z]HE:Z]Z
x
+
a,BC^WF#'()*+#gKZO./+#:;
a/ f(x) =
H
H E:HZ]H:^Z]Zc
biết F(
^
π
)= - 3
2
b/ f(x) =
^ H
H
x x
x
− −
biết F((-3) = 10
HXH,'()*+#6'47'4489:;
12YVW&$%=49'&$%9!

>$%?@=") !8
A
 \K(OB([ K OF u C+
∫
5
 \l(KZOnBZ[ l K OnF u x C+
∫
Ví dụ1W,'()*+#./+#:;:/(
/
E:Z
:e xdx
∫

f
Kf ^Ox dx+
∫

^
H
E:
tg x
dx
x
∫
B
:Z
E:e xdx
∫
Ví dụ 2 W,
/

H
^
q
dZ
x
dx
∫

H
`
x x dx−
∫

H

K O
dx
x x+
∫
B


x
dx
e +
∫
Bài tập1W,
/
f
Z

: E:
^ ^
x
c dx
∫

^ H
`
x x dx−
∫

H


dx
x x−
∫
B
HY ^x
dx
x
+
∫
Bài tập2W,
/
 /Zdx
∫

E Zdx
∫


^
K/  /ZOdx+
∫
B
^
KE Z]E ZOdx
∫
g/
H

r q
dx
x x− +
∫
h/
H
^ H
r q
x
dx
x x
+
+ +
∫
HXX^,'()*+#6'47'44='4!
12YVW3"8'!95
 (KZO<sKZOBZ[ K O K O K O sK Ou x v x v x u x du−
∫ ∫
Ví dụ 1W,'()*+#./+#:;:/(

/\KZO[Z
H
:HZ\KZO[Z
H
E:Z
\KZO[Z
H
e
Z
B\KZO[Z
^
YKHZO
Ví dụ 2W,
a/
:Zx dx
∫
b/
Z
K Ox e dx+
∫
c/
KH O:x xdx+
∫
(x) d/
`
: E:x xdx
∫
e/
f
Kf ^Ox dx+

∫
g/
Z
e :Zdx
∫
h/
Yx xdx
∫
k/
H
Yx xdx
∫
Bài tập 1W,
a/
H
:Zx dx
∫
b/
dZ
xe dx
∫
c/
:
^
x
x dx
∫
(x) d/
: H
x

e xdx
∫
Bài tập 2:,
a/
H x
e dx
+
∫
b/
H
/x xdx
∫
3
c/
E:KYZOc dx
∫
(x) d/
H
YK Ox x dx+
∫
HXHX,,456':MBC',-<+'()*+# 
Ví dụ 1W,
a/

^
_
K Ox x dx+ +
∫
b/
H

H

 
K O
e
x x dx
x x
+ + +
∫

c/
^

Hx dx−
∫
d/
H

x dx+
∫

Ví dụ 2W,
a/
H
^
KH: ^ Ox cosx x dx
π
π
+ +
∫

b/

_
K O
x
e x dx+
∫

c/

^
_
K Ox x x dx+
∫
d/
H

K OK Ox x x dx+ − +
∫

Bài tập W,
a/
H
^

K^: H Ox cosx dx
x
π
π
+ +

∫
b/

H
_
K O
x
e x dx+ +
∫

c/
H
H
^

K Ox x x x dx+ +
∫
d/
H

K OK Ox x x dx− + +
∫

HXHXH,,456'47'4489:;
o'B9:;
K O
K O
l(K On(sKZO Z K O
u b
b

a u a
f x d f u du=
∫ ∫
Ví dụ1W,
a/
H
^ H
^
: xcos xdx
π
π
∫
b/
H
H ^
^
: xcos xdx
π
π
∫

c/
H
_
:
 ^
x
dx
cosx
π

+
∫
d/
`
_
tgxdx
π
∫

g/
`
r
E gxdx
π
π
∫
h/
r
_
 `: xcosxdx
π
+
∫
Ví dụ 2W,
a/

H
_
x x dx+
∫

b/

H
_
x x dx−
∫

c/

^ H
_
x x dx+
∫
d/

H
^
_

x
dx
x +
∫

g/

^ H
_
x x dx−
∫

h/
H
^



dx
x x +
∫
Ví dụ 3W,
4
a/

H
_


dx
x+
∫
b/

H


H H
dx
x x
−
+ +

∫

c/

H
_


dx
x +
∫
d/

H H
_

K ^ O
dx
x+
∫
g/
H
:
`
x
e cosxdx
π
π
∫
h/

H
`
:
cosx
e xdx
π
π
∫

f/
H

H
_
x
e xdx
+
∫
k/
H
^ H
^
: xcos xdx
π
π
∫

Bài tập 1W,
1/
H

:
`
x
e cosxdx
π
π
∫
2/
H
`
:
cosx
e xdx
π
π
∫
3/
H

H
_
x
e xdx
+
∫
Bài tập 2W,
1/
H
^ H
^

: xcos xdx
π
π
∫
2/
H
H ^
^
: xcos xdx
π
π
∫
3/
H
_
:
 ^
x
dx
cosx
π
+
∫

4/
`
_
tgxdx
π
∫

5/
`
r
E gxdx
π
π
∫
6/
r
_
 `: xcosxdx
π
+
∫
Bài tập 3W,
1/

H
_
x x dx+
∫
2/

H
_
x x dx−
∫
3/

^ H

_
x x dx+
∫
4/

H
^
_

x
dx
x +
∫
5/

^ H
_
x x dx−
∫
6/
H
^



dx
x x +
∫
Bài tập 4 W,
1/


 Y
e
x
dx
x
+
∫
2/

:KY O
e
x
dx
x
∫
3/

 ^Y Y
e
x x
dx
x
+
∫
4/
HY 

e
x

e
dx
x
+
∫
5/
H
H
 Y
Y
e
e
x
dx
x x
+
∫
6/
H
H

K Y O
e
e
dx
cos x+
∫
Bài tập 5W,
1/
H


 
x
dx
x+ −
∫
2/

_
H 
x
dx
x +
∫
3/

_
x x dx+
∫
4/

_


dx
x x+ +
∫
5/

_



dx
x x+ −
∫
6/
^

x
dx
x
+
∫

HXHX^,,456'47'44='4!
o',45='4!W
(K O<sKZO Z K O K O K O sK O
b b
b
a
a a
x d u x v x v x u x dx= −
∫ ∫
5
BCDEFG'E'
tDng 1
:
K O
ax
ax

f x cosax dx
e
β
α
 
 
 
 
 
∫

K O sK O
: :
E:
ax ax
u f x du f x dx
ax ax
dv ax dx v cosax dx
e e
= =
 
 
   
 
⇒
 
   
= =
 
   

 
   
   
 
∫
tDng 2:
K OYK Of x ax dx
β
α
∫
1u
YK O
K O
K O
dx
du
u ax
x
dv f x dx
v f x dx

=
=


⇒
 
=



=

∫
Ví dụ1W,
/
^
^

Y
e
x
dx
x
∫


Y
e
x xdx
∫


H
_
YK Ox x dx
+
∫
B
H


Y
e
x xdx
∫

Ví dụ 2W,
/
^
^

Y
e
x
dx
x
∫


Y
e
x xdx
∫


H
_
YK Ox x dx
+
∫
B

H

Y
e
x xdx
∫
Tích phân tng phn cc hàm s cn kh!o l!o đ%t u và dv
a,BCHW,,45:/(
/

H
H
_
K O
x
x e
dx
x +
∫
8u
H
H
K O
x
u x e
dx
dv
x

=



=

+


^
v
` ^
H
K O
x dx
x −
∫
8u
f
^
` ^
K O
u x
x dx
dv
x

=


=


−


   
H H H
 H
H H H H H H H
_ _ _ _

K O K O  K O
dx x x dx x dx
dx I I
x x x x
+ −
= = − = −
+ + + +
∫ ∫ ∫ ∫
,


H
_

dx
x
=
+
∫
6'47'4489:;
,

H
[

H
H H
_
K O
x dx
x+
∫
6'47'44='4!W8u
H H
K O
u x
x
dv dx
x
=



=

+

6
Ví dụ 2WF#Zw_:/EE

H
H

_

K O
x
t e
dx
t
=
+
∫
0+S4W,,45:/(
/
H
_
K E:ZO:Zx c dx
π
+
∫



K OY
e
x xdx
x
+
∫

H
H


YK Ox x dx
+
∫
B
^
H
`
/x xdx
π
π
∫
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Công thức tính diện tích hình phẳng
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K O Z
b
a
f x d
∫
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x x
b b
a a x x
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j
i
x
x
f x d
∫
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d
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Z
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^
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π
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Z
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^
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π
tE/+#:;)[\KZON)['KZOY*CD*8EJl/mnNB&,F4R''bJx
)[\KZON)['KZO<+/87z'R'Z[/NZ[87i,eEo':/(
L[
K O K O Z
b
a
f x g x d−
∫
t,B&,F4R''bJx)[\KZON)['KZO<+/87z'R'Z[/NZ[
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H

8
T
8@@@8
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7
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 H

K O K O Z K O K O Z] K O K O Z]XXX K O K O Z
n
x x
b b
a a x x
f x g x d f x g x d f x g x d f x g x d− = − − + −
∫ ∫ ∫ ∫
P(VQB.4>
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K O K O Z
j
i
x
x
f x g x d−
∫
HX^XHP4BC',45,Q,<SQ

8